THEORY OF DETERMINANTS Lontion: C. J. CLAY AND SONS, CAMBRIDGE UNIVERSITY PRESS WAREHOUSE, AVE MARIA LANE. %gOIau5o: 50, WELLINGTON STREET..Leipig: F. A. BROCKHAUS. e CJUorkt: THE MACMILLAN COMPANY. Bomba aib Calcutta: MACMILLAN AND CO., LTD. [AEl RVig/ts reserved. THE THEORY OF DETERMINANTS AND THEIR APPLICATIONS BY ROBERT FORSYTH SCOTT, M.A. OF LINCOLN'S INN; FELLOW OF ST JOHN'S COLLEGE, CAIMBRIDGE. SECOND EDITION: REFISED BT G. B. MATHEWS, M.A., F.R.S. FELLOW AND LECTURER OF ST JOHN'S COLLEGE, CAMBRIDGE; UNIVERSITY LECTURER IN MATHEMATICS. CAMBRIDGE at the University Press 1904 cambrigr: PRINTED BY J. AND C. F. CLAY, AT THE UNIVERSITY PRESS. PREFACE TO THE FIRST EDITION. N the present treatise I have attempted to give an exposition of the Theory of Determinants and their more important applications. In every case where it was possible I have consulted the original works and memoirs on the subject; a list of those I have been able to see is appended as it may be useful to others pursuing the same line of study. At one time I hoped to make this list exhaustive, supplementing my own researches from the literary notices in foreign mathematical journals, but even with this aid I found that it would be necessarily incomplete. In consequence of this the list has been restricted to those memoirs which I have seen, the leading results of which are incorporated either in the body of the text or in the examples. The principal novelty of the treatise lies in the systematic use of Grassmann's alternate units, by means of which the study of determinants is, I believe, much simplified. I have to thank my friend Mr JAS. BARNARD, M.A. of St John ' College and Mathematical Master at the Proprietary School, Blackheath, for the care he has bestowed on correcting the proofs and for many valuable suggestions. R. F. SCOTT. February/ 1880. a3 REVISER'S PREFACE. THE principal changes made in this edition are that some account has been given of infinite determinants, and of the elements of the theory of bilinear forms, together with the fundamental propositions about elementary divisors. I have intentionally refrained, as far as possible, from altering the character of the book, or increasing its size. The list of books and memoirs relating to determinants has been omitted, Dr Muir's bibliography being easily accessible; instead of this I have given a brief account of the earlier history of the subject. The new introductory chapter is intended for beginners, who are apt to feel discouraged if they first approach the theory in its most general form. For a similar reason the abbreviated notation employed in some chapters has not been used in those which are more elementary. Besides original papers, I have consulted Pascal's excellent treatise in the Hoepli series, and Muth's Elemnentartheiler. The first volume of Kronecker's lectures on determinants appeared too late for me to consult it. I ought to say that for all the changes that have been made I am solely responsible, the revision having been left entirely in my hands by the author. G. B. MATHEWS. May 1904. CONTENTS. CHAPTER I. INTRODUCTION. ART. PAGE 1, 2. Notation. Linear equations.... 1 3. Rule of Sarrus........ 3 4-6. Elementary properties of determinants of the third order. 4 7, 8. Product of two determinants...... 6 CHAPTER II. DEFINITIONS AND NOTATION. ALTERNATE NUMBERS. 1-7. Permutations of Elements...... 8 8-10. General definition of a Determinant.... 13 11. Interchange of rows and columns..... 16 12-14. Alternate numbers........ 16 15. Expression of a determinant as a product of alternate numbers........ 18 16, 17. Examples......... 19 CHAPTER III. GENERAL PROPERTIES OF DETERMINANTS. 1, 2. Interchange of rows and columns..... 22 3, 4. Value of a Determinant when the elements of a row are sums 23 5, 6. Examples.......... 24 7. Solution of a system of linear equations... 26 viii CONTENTS CHAPTER IV. ON THE MINORS AND ON THE EXPANSION OF A DETERMINANT. ART. PAGE 1. Number of Minors of order p. o. 28 2-4. Complementary Minors...... 28 5. Laplace's Theorem........ 30 6, 7. Examples.......... 32 9-11. Expansion of a Determinant according to the elements of a row.......... 34 12-14. Examples......... 37 15, 16. Differential Coefficients of Determinants.... 39 18-21. Albeggiani's expansion of a Determinant with polynomial elements......... 42 22, 23. Expansion of a Determinant according to products of elements in the leading diagonal. 44 24. Cauchy's theorem......... 47 25. Examples........ 47 CHAPTER V. COMPOSITION OF ARRAYS. MULTIPLICATION OF DETERMINANTS. 1-4. Determinant formed by compounding two arrays.. 49 5-7. Examples.......... 52 8. Fundamental theorem deduced from Laplace's theorem. 56 9. Minor of a product Determinant...... 57 10. Differential Coefficient of a product Determinant. 58 CHAPTER VI. ON DETERMINANTS OF COMPOUND SYSTEMS. 1-3. Reciprocal Arrays....... 60 4, 5. Reciprocal Arrays of the first order... 61 6, 7. Examples........ 63 8-11. Reciprocal Arrays of the mth order.... 65 12-16. Theorems of Sylvester, etc........ 67 17. Theorem of Netto........ 70 18, 19. Theorems of Kronecker.... 71 20, 21. Case of two independent systems... 73 CONTENTS ix CHAPTER VII. ARITHMETICAL PROPERTIES OF DETERMINANTS. ELEMENTARY FACTORS. ART. PAGE 1. General determinant irreducible...... 75 2-4. Determinant factors and elementary factors defined.. 76 5-8. Regular minors. Properties of elementary factors. 77 9. Equivalent matrices....... 82 10-13. Reduction to normal form..... 83 CHAPTER VIII. DETERMINANTS OF SPECIAL FORMS. 1-3. Symmetrical Determinants....... 88 4-8. Skew and skew symmetrical Determinants.... 90 9-16. Skew symmetrical Determinants, Pfaffians.... 92 17, 18. Skew Determinants, Examples...... 97 20-22. Orthosymmetrical Determinants...... 99 23-26. Determinants whose elements are arranged cyclically.. 102 27-30. Determinants whose elements are binomial coefficients. 104 CHAPTER IX. CUBIC DETERMINANTS, AND DETERMINANTS WITH MULTIPLE SUFFIXES. 1, 2. Definition, Notation........ 110 3. Expression for a cubic determinant as a product of alternate numbers......... 111 4-9. Elementary properties of cubic determinants... 11 10-18. Determinants with multiple suffixes.... 114 19, 20. Examples.......... 118 CHAPTER X. DETERMINANTS OF INFINITE ORDER. 1-4. Definitions......... 120 5-9. Properties of normal determinants... 122 10. Product theoremn....... 126 11, 12. Seminormal determinants....127 x CONTENTS APPLICATIONS. CHAPTER XI. APPLICATIONS TO THE THEORY OF EQUATIONS AND OF ELIMINATION. ART. PAGE 1-6. System of Linear equations...... 130 7-9. Linear substitutions....... 133 10-15. Resultants and discriminants...... 135 16. Property of Bézout's determinant..... 142 17, 18. Quadric and linear system... 144 19. Reality of the roots of the equation for the secular inequalities 147 CHAPTER XII. RATIONAL FUNCTIONAL DETERMINANTS. 1-3. Product of all the differences of n numbers...149 4-17. Examples of functional determinants. 150 CHAPTER XIII. JACOBIANS AND HESSIANS. 1. Definition and notation for a Jacobian....163 2. The Jacobian of dependent functions vanishes.. 163 3-5. Jacobian of functions with a common factor...164 6-10. Properties of Jacobians...165 11. Bertrand's definition........ 170 12. Definition bv means of alternate numbers.. 171 13, 14. Transformation of a multiple integral.... 172 15. Definition of a Hessian. Example..... 176 16. Jacobians and Hessians are covariants... 177 17. Jacobian of n linear functions. Hessian of a quadric. 179 CHAPTER XIV. APPLICATIONS TO BILINEAR AND QUADRATIC FORMS. 1-3. Symbolical multiplication of bilinear forms... 180 4. Characteristic equation....... 182 5-7. Applications of symbolical calculus.. 182 8, 9. Frobenius's proof of Kronecker's theorem....186 10-15. Reduction of bilinear and quadratic forms.... 189 16, 17. Simultaneous reduction of two quadrics... 194 18-24. Orthogonal substitutions.......196 25. Invariant of system of quadrics expressed as a cubic determinant....... 202 CONTENTS xi CHAPTER XV. DETERMINANTS OF FUNCTIONS OF THE SAME VARIABLE. ART. PAGE 1-9. Definition, Elementary properties..... 203 10. Application to linear differential equations.... 209 11. Hesse's Solution of Jacobi's equation.... 210 CHAPTER XVI. CONTINUED FRACTIONS. 1. Ascending and descending Continued Fractions... 212 2-7. Expression for the Convergents to descending Continued Fractions. Elementary properties.....213 8, 9. Ascending Continued Fractions, Transformation to descending Continued Fractions.......216 10. Conversion of a series into a continued fraction... 219 11. Fiirstenau's extended Continued Fractions.... 220 CHAPTER XVII. APPLICATIONS TO GEOMETRY. 1-3. Area of a triangle, volume of a tetrahedron...223 4-6. Elementary identities....... 226 7. Application of alternate numbers..... 228 8-14. Angles between straight lines, Solid angles, Spherical figures 230 15-18. Systems of straight lines, Co-ordinates of a line, Mutual moments........ 236 19-23. Relation between the lines joining five points in space (Cayley), Volumes of tetrahedra, Areas of triangles, Siebeck's theorem........ 239 24 —28. Formulse relating to an Ellipsoid, Theorem of Cayley for six points on a Sphere, Faure's theory of Indices 244 29-39. Systems of spheres, Mutual powers, Common tangents.249 EXAMPLES............ 257 HISTORICAL NOTE........ 286 THEORY OF DETERMINANTS. CHAPTER I. INTRODUCTION. 1. DETERMINANTS are algebraical expressions of a particular type calculated by a systematic rule and expressed by a special notation. Accordingly there is a calculus of determinants; and besides this there is a theory, dealing with the properties of determinants which result from their analytical form. The most natural way of introducing the subject is to consider a few simple cases of the problem to which the invention of determinants is due; namely, the formal solution of a general system of simultaneous linear equations. If the system is ax + a2y + a3 = 0 bx + b2y + b3 = 0 the solution is at once found to be a2 b3 - ab2 a3b - alb3 alb2 -a2b-' alb2 - a2b1' and in the same way the homogeneous equations alx + ay + a3z = 0} blx + b2y + b3z = Oj lead to the proportion x: y: z= (a2b- a3b2): (a3b -ab3): (ab2-ab,), from which the previous result follows by putting z = 1. S. D. 1 2 THEORY OF DETERMINANTS [CHAP. I. Consider next the homogeneous system ul ax ay ax + ay a t = 0O u2 - b1x + b2y + bz + b4t = 0.. u3 = clx + c2y+c+ + + c4t = O In the derived equation X1u1 + X2u2 + X32t3 = 0 the coefficients of z and t will vanish if Xia3 + X2b3 + X3c3= O, X a4 + X2b4 + 3C4-=; that is, by the preceding case, if X: Xa: X3= (b3C4- b4C3): (c3a4 -c4a) (a3b4-a4b3). Taking the multipliers in this proportion, the derived equation becomes Qx + Py = 0, where P = a2 (b3c4 - b4c3) + b2 (ca4 - ca3) + c2 (a.b4- a4b), Q = a, (b3c4 - b4c3) + b1 (c3a4 - c4a) + ci (a3b4 - a4b3). The expressions P, Q are of precisely similar form, and differ only in the sets of coefficients which they involve. It is convenient to write a2 a3 a4 P= b2 b3 b4 C2 C3 C4 in this notation P is said to be expressed as a determinant of the third order. The determinant has three rows, such as a2, a,, a4; three columns, such as a2, b2, c2; and nine elements, a2, a3,... c4. With the same notation ai a3 a4 Q= b, b3 b4 Cl C3 C4 1-3] INTRODUCTION 3 and it is found without difficulty that the complete solution of the given system is x -y z -t a2 a3 a4 ai a.3 a4 al a2 a4 a1 a2 a3 b2 b3 b4 b1 b3 b4 b1 b2 b4 b1 b2 b3 C2 C3 C4 C1 C3 C4 C1 C2 C4 C1 C2 C3 2. We may use this result to solve, by the method of multipliers, a system of four homogeneous equations in five unknowns, and so on. The general result, as will be proved later on, is that (n +1) variables satisfying n linear homogeneous equations are proportional to (n + 1) rational integral functions of the coefficients, each homogeneous and of dimension n in a certain set of n2 coefficients. These expressions are, in fact, determinants of the nth order, analogous to those of the third order above defined. A determinant of the first order, la, is merely the single term a; for the second order, we have ai a2; b, b ~ Iab, - abbl; b1 b2 the general rules for expanding a determinant of any order will be found in the next chapter. 3. A convenient rule for writing down the expansion of any determinant of the third order is the following, due to Sarrus. Let the determinant be ai a2 a3 bi b2 b3 C1 C2 CS Alongside of this repeat the first and second columns in order aQ a2 a3 a, a2 b1 2 b b3 >' b2 C C2 C3 C1 C2 ana form the product of each set of three elements lying in lines 1-2 4 TIEORY OF DETERMINANTS [CHAP. I. parallel to the diagonals of the original square. Those which lie in lines descending from left to right have the positive, the others the negative sign. Thus the determinant is alb2c + asb3cl + a3bc2 - c3b2cl - ai b3ca - a2 b c. In practice it is not necessary actually to repeat the columns, but only to imagine them repeated. 4. In order to make the notation familiar, proofs will now be given of some elementary properties of determinants of the third order. They are special cases of theorems which are true for any order; and the reader will easily verify their for determinants of the second order. Consider the determinant al a2 a3 D = bl b b3 = a b2c3-aib3c + a2b3c-a2bc3 + a.3bc2- ab,2c. Cl C2 C3 Each term of the expansion is of the forni + aab c,, where (a, 3f, ry) is a permutation of (1, 2, 3); in other words, it is a product of three elements, no two of which belong to the same row or to the same column. The sign of the term alb2c3, derived from the principal diagonal (that which slopes down from left to right), is positive: any other term abc, is preceded by the sign - or + according as (a, /, y) is derivable from (1, 2, 3) by a single transposition or two transpositions. Thus half the terms are positive, and half negative. The value of D is unaltered if columns are changed into rows without altering the positions of a1, b2, c3: that is to say, a1 a2 a 1 al b1 cl bl b2 b3 = a2 b2 C2. C1 C2 3 a3 b3 C3 This is verified by expanding each side. 5. If any two rows or columns of D are interchanged, the value of the new determinant is - D. For the effect of the 3-6] INTRODUCT1ON 5 change on the expansion of D is either to change two letters (such as a, b) without altering the positions of the suffixes, or else to interchange two suffixes: in each case the result is - D. For instance, if the second and third rows change places, the new determinant is ai a1 a2 C1 c2 c3 = alc2bs - alcb2 + a2c3b - a2clbs + a3 cb2 - a3c2b, b1 b2 b3 = — D. Hence if two rows or two columns are identical, the value of D is zero. 6. We have D = a (b2c3 - bc2) + a2 (b31 - blc3) + a3 (bic2 - b2c1) b2 b3 b 6b bi b.2 -a= a + a; C2 C3 3 CI C1 C2 thus D is a linear homogeneous function of the elements ai, a2, a. in the first row, the coefficients being determinants of the second order constructed from elements in the other rows. D can be similarly expressed as a linear function of the elements of any other row or columnn. It follows from this that pl + ql, p2 +, P3 +q3 Pi p2 Ps qi q2 qS bi, b2, b3 bi b2 b3 + bi b2 b3 C1 c 2, c3 c1 C2 C3 C1 C2 C3 and there is a corresponding theorem when the elements of any column are binoinials. More generally, if each elelnent of the first column is expressed as the sum of I terns, each of the second as the sum of m terms, and each of the third as the sum of n terms, then D can be expressed as the sum of Imn determinants, in each of which the columns consist of corresponding terms of the sums in question. There is, of course, an analogous theorem with " rows" instead of "columns." 6 THEORY OF DETERMINANTS [CHAP. I. Again, if the elements of any row or column are multiplied by k, the value of D is multiplied by k. For instance, ka, a a3 a1 a2 a3 kb1 b2 b3 = k x bl b2 b3 kcl C2 C3 C1 C2 C3 By combining the foregoing results, we obtain the useful theorem that ka + la2 + ma3, a, a3 al a2 a kb~ +1 b, + mb3, b2, b3 =k x b1 b2 b3 = kD. kcC + IC2 + nc3,, c3, C3 1 C 2 c3 This follows froin the fact that the determinant on the left can be expressed in the form k^ +lA,+mA3, where A =D, and the determinants A,, A3 vanish on account of the identity of two columns. 7. The product of two determinants of the third order can be expressed as a determinant of the third order. To see this, consider the determinant aa + b/ + cy, aa' +-b/' +cy', aaci+b3" ~ cy" D= a 'a + b'3 +c'y a' ' + b'3' + c'y', a'a" + b'" c'y" a"a + b"l3+ c",/, a"' +b"/3' +c"y', a"a"+ b"3"+c"/" By selecting partial columns, this can be expressed as the sum of 27 determinants such as aa aa' b/3" ac b/3' cy" a'a a't b'3", a'a b',3' c'y" a"a a"a' b"f3" a"0 b"a' c"y" and so on. But of these the only ones that do not vanish are the six which, like the second one above written, contain all the nine elements of the determinant a b c a' b' c. a" b/ c" | C 0 6-8] INTRODUCTION 7 Calling this d, and putting a 3 7 a,' 3' ' =8, a",i" 7" each of the set of 27 which does not vanish is the product of d by a term of 8; for instance, the one above given is equal to a/3'/y"d. Moreover it is found, on examination, that the sign of the term which multiplies d agrees with the sign which it has in the expansion of 8; and that every term of 8 is represented in this way. Consequently D =dS. 8. In expressing the product dS as above, the multiplication is said to be effected by rows; in fact each element of D is the sum of products of corresponding pairs of elements of rows in d and 3. Since the values of d, 8 are not affected by interchanging rows and columns, there are four ways of performing the multiplication: the resulting determinants are, in general, distinct in form, though their complete expansions are, of course, the same identically. To illustrate the different methods of procedure, we may take the case of two determinants of the second order. By the same kind of reasoning as before, it is verified that each of the determinants a' + b/3, a'a' +b' a'a+ b'', a/3 + +b3' aa + b'/3, a' + b'3' a/a + bC'a, a'/3 + b',S' aa + a a'', a + a'/' aa + a'a', a3 + a'f' ba + b'fi, ba' + b'fi' ba + b'a', b/3 + b'i3' a b a, is equal to the product of a' b and a' a' b' Ct /3' CHAPTER II. DEFINITIONS AND NOTATION. ALTERNATE NUMBERS. 1. BEFORE proceeding to the theory of determinants of any order, it is convenient to recall a few theorems relating to the permutations of n different things in a line. If we have any n elements al, a2,... a., we may call a,, a2,.. a., where the elements are arranged according to the magnitude of the numbers forming the suffixes, the natural or original order of the letters. Any other order is called a permutation of the elements. One element is said to be higher than another when it has the greater suffix. When in any permutation an element with a higher suffix precedes another with a lower, we have an inversion. Thus the permutation a4, a2, ai, a., of four letters, contains the following four inversions, a4 a2, a4a1, a4a, a 2ai, where we compare each element with all that follow it. Following Cramer it is usual to divide the permutations of a given set of elements into two classes; the first class contains those permutations which have an even number of inversions, the second those which have an odd number. 1-2] DEFINITIONS AND NOTATION 9 2. By permuting the elements a,, a2,... a, we obtain all possible ways in which they can be written. The same result is arrived at by writing down all the permutations of the suffixes 1, 2,... n and then putting a's above them. By repeated interchange of two suffixes we can get every permutation of the given elements from their original order. For if we start with two suffixes 1, 2, they have but two arrangements, 1, 2, 2, 1, of which the second is got from the first by a simple interchange. Taking three elements 1, 2, 3, out of these we can select the duad 2, 3, whose permutations are 2, 3; 3, 2. Prefixing 1 to each of these we get 1, 2, 3; 1, 3, 2, which are two permutations of the given elements. Proceeding in like manner with the other duads 1, 3; 1, 2, we get the six arrangements of three figures 123, 1 3 2, 231 213, 312, 321. Next take four numbers 1, 2, 3, 4. We get four triplets by leaving out one number, viz. 123, 124, 1 34, 234. For each triplet we can write down six arrangements by the rule just given for three numbers, then adding on the missing number we get twenty-four arrangements of four numbers, viz. 1234 1243 1342 2341 2134 2143 3142 3241 1324 1423 1432 2431 3124 4123 4132 4231 2314 2413 3412 3421 3214 4213 4312 4321. And so we could go on to write down the arrangements of any set of elements. The number of arrangements of n letters is 1. 2. 3... n or n!, an even number. 10 THEORY OF DETERMINANTS [CHAP. II. 3. If in a given permutation two elements be interchanged while all the others remain unaltered in position, the two resulting permutations belong to different classes. This will be proved if we can shew that the difference between the number of inversions in the two permutations is an odd number. We can represent any permutation of a group of elements by A d B e C............................(1), where d and e are the two elements to be presently interchanged, A the group of elements which precede d, B the group between d and e, and C the group which follows e. The permutation we obtain is A e B d C............................(2). The number of inversions in the two permutations (1) and (2) due to the elements contained in the groups A, B and C is in each case the same. And since the elements of A precede d and e in both permutations we get no new inversions in (2) from these; the elements of C follow both d and e, and therefore give rise to no new inversions. We have therefore only to consider the changes in the two permutations d B e and e B d......................(3). Suppose that e is higher than d; let B contain b elements of which b, are higher than d and b2 higher than e. Then in the permutation d B e we have, independently of the inversions contained in B itself, b - b, + b, inversions, because there are b - b, elements lower than d and b2 higher than e. In e B d we have b- b2 inversions on account of e, b1 on account of d, and one because e is higher than d; thus, without counting the inversions in B, we have b - b2+ b,+ 1. The difference between the number of inversions in the permutations (3), and therefore in (1) and (2), is thus b-b2b+ i+1 -(b-b1b2) = 2(b- b2)+ 1, which is an odd number, shewing that the permutations belong to different classes. 3-5] DEFINITIONS AND NOTATION 11 4. The same result may be arrived at as follows. If there be n quantities whose natural order is a1, a2,... an, and if in any arrangement we subtract each suffix from all that follow it and multiply these differences together, we shall have a product whose sign will depend on the number of inversions in the given arrangement, the sign being positive if the number of inversions is even and negative if the number of inversions is odd. If then i, k be any two suffixes chosen arbitrarily which are to be interchanged, i preceding k in the given arrangement, the product of the differences will consist of four parts. (i) The factor k- i. (ii) and (iii). A set of factors such as + (r - k), and + (r - i), where r is some number of the series 1... n excluding i and k. (iv) A set of factors such as r - s, where r, s are two numbers of the series 1, 2,... n excluding i and k. Then for the given arrangement the product of the differences will be (k -i) n (r - i) (r-k) (r- s), where e denotes + 1 or - 1 as the case may be. If now we interchange i and k, the signs of all factors such as (r-k)(r-i), (r-s) remain unchanged, while k-i changes sign. Thus on interchanging two elements the product of the differences changes sign, i.e. by interchanging two suffixes we have introduced an odd number of negative factors and therefore of inversions, hence the two arrangements considered belong to different classes. 5. If in a series of elements each is replaced by the one which follows it, and the last by the first, we are said to have got a cyclical permutation of the given arrangement. If the system of elements al, a2,... an be considered as forming an endless band, if we cut this band between a, and an we have the natural order, cutting it between 12 THEORY OF DETERMINANTS [CHAP. II. a, and a2 we have a cyclical permutation of the first order, and so on. Such a cyclical permutation is equivalent to n - 1 simple interchanges, viz. we move a, from the first to the last place by interchanging the first and second elements, then the second and third, and so on, in all n- 1 simple interchanges. Thus a cyclical permutation of a given arrangement belongs to the sane or opposite class as the given one according as the number of elements is odd or even. 6. Every permutation of a given set of elements may be considered as derived from a fixed permutation by means of cyclical permutations of groups of the elements. This is best illustrated by an example. Let the suffixes of two permutations of nine elements be 7, 6, 3, 2, 1, 4, 8, 5, 9 8, 7, 9, 5, 1, 6, 4,?, 2 To obtain the second permutation from the first, we begin by replacing 7 by 8, 8 by 4, 4 by 6 and 6 by 7, which completes a cycle. Then we replace 3 by 9, 9 by 2, 2 by 5 and 5 by 3, which completes another cycle. Lastly, 1 forms a cycle by itself. 7. If elements which remain unchanged like 1 in the preceding example be considered as forming a cycle of one letter, we may state the following theorem: Two permutations belong to the sanie or different classes, according as the difference between the number of elements and the number of groups by whose cyclical interchange one permutation is got from the other, is even or odd. For if there be n elements altogether, and p cycles of n1, n2... n. letters respectively, the cyclical interchanges are equivalent to (ni - 1) + (n2- 1) +... + (np- 1)= n ++... +np-p == n - p simple interchanges, which proves the theorem. In the example in Art. 6, n= 9, p = 3, and thus the permutations belong to the same class. 5-8] DEFINITIONS AND NOTATION 13 8. A determinant of the nth order is a function of n2 elements, which are conveniently distinguished by double suffixes in the following manner. The complete symbol for the determinant is written ai, a2... an a1 ai2... a a,, an2 ann so that the element ars is the sth constituent of the rth row, or (which is the same thing) the rth constituent of the sth column; rows being counted from above downwards, and columns from left to right. The elements may be regarded as being arranged in a square block; of this the line containing an, a2,... ann is called the leading or principal diagonal. The actual block of elements, considered per se, is called an array, or inatrix. More exactly, it is a square array, as distinguished from other arrays. The expansion of the determinant represented by the above symbol is obtained by the following rule: From the array choose n different elements such that there is one and only one element from each row and column, and multiply these elements together; the product will be a term of the determinant. For example, the product of the elements an, a22,... a.nn situated in the principal diagonal of the square array, is a term of the determinant; this will be called the leading term, and to it we attribute the positive sign. The sign of any other term Cfg. ahk.. ast is determined as follows. From the mode in which the elements were selected, it follows that f, h,...s, and g, k,...t are each of them permutations of 1, 2,... n. Let them contain p and q inversions respectively, then the sign of the term afg. ahk... ast is (- 1)P+q. The sum of all the possible terms with their proper signs is the determinant of the array. 14 THEORY OF DETERMINANTS [CHAP. II. More simple rules may be given for determining the sign of any term. If we interchange any two elelnents ahk and aij the term does not change its sign. For this interchange is equivalent to the interchange of i with h and j with k. By these two interchanges we increase both p and q by an odd number, and hence the sign of the term is unaltered. It is therefore usual to give to one series of suffixes their natural order, so that one of the two numbers p or q becomes zero, and the sign of the term of the determinant now depends solely on the number of inversions in the other series, and is the same whether the first or second series of suffixes retains its natural order. It is thus clear that all the terms of the determinant will be obtained from the leading term ani a'.2.. ann by keeping the first suffixes fixed in their natural order, and writing for the second suffixes in succession all possible permutations of the elements 1, 2,... n, giving to the product of the elements the positive or negative sign according as the number of inversions is even or odd. Such a determinant is said to be of the nth degree, since each term is the product of n elements. It has n! terns in all, since this is the number of permutations of the second suffixes, each of which gives a term of the determinant. Half of these terms have the positive, the rest the negative sign. 9. Various notations are employed for the determinant of a system of n2 elements. Cauchy and Jacobi denoted it by drawing two vertical lines at the sides of the array, or by writing + before the leading term and prefixing a summation sign, an, al2,... an = a... ann' a21, a, n,... a2..................... ani an2, * * ann \ Sylvester uses the umbral notation 1, 2,... n, 1, 2,... n. 8-10] DEFINITIONS AND NOTATION 15 If the determinant be written in the forin XI 1, Yi, Z1, ~ 2 y2 Y2 Z2 xIn /Yn Zn... we may denote it by Ixi, yi, Zi,,... (=l 1, 2,... n), meaning by this that i is to take the different values 1, 2,... n in succession. Lastly, the determinant with double suffixes may be denoted by \ aik (i, k =l, 2,... n), the bracket at the side telling us what values the suffixes i and k take. This notation is, I believe, due to H. J. S. Smith, who employs it in his report on the theory of numbers, Brit. Ass. Rep., 1861, p. 504. It clashes with Weierstrass's use of the symbol z to denote the absolute value of the complex quantity z; but this does not lead practically to any confusion. If we use the symbol, ann i, the explanatory bracket becomes unnecessary, since the notation now indicates the order of the determinant, and it is, of course, understood that the elements unexpressed are of the type ars, with r n, s n. 10. Illustrations of the rule for expanding a determinant will be found in the introductory chapter. It may be observed that when the elements are represented by literal symbols with single suffixes, it is usual to write the factors of any term in the dictionary order of the letters: its sign is then determined by the class of the permutation of the suffixes. For instance, a1 b c1 d= ` a bbc3d4 - a2bac3d4- ab3c2d4 + a3blc,d4 a2 b2 c2 d2 + a2bcld4 - ab2cld4 - albc4d3 + a2bc4d3 a3 b3 C3 d3 + alb4c2d3- a4blc2d3s- a2b4cld + a4b2cld3 a4 b4 c4 d4 + albc4d2a - ablc4d2 - alb4c3d2 + a4blc3d2 + a3b4cId2 - a4b3cld2 - a2b3c4dl + a3b2c4dl + a2b4c3dl - a4b2c3d - a3b4c2dl + a4b3c2di. 16 THEORY OF DETERMINANTS [CHAP. II. In the double-suffix notation, the same expansion is, in the same order, a44 |1= -a,, a,, 4 a,, a12a, a44 - ~ a4 aa a2a23a4 + etc. 11. If we interchange rows and columns in the determinant of Art. 9, we get all, a21, ~* an,, a12, a22).. afn2.................. ain, a2,,.. ann This is the same as the original determinant with the suffixes of each element interchanged. Its expansion is then obtained from that of the original determinant by interchanging in each term the suffixes of each element. That is to say, in the term ala22... ann we keep the second suffixes fixed in their natural order and write for the first suffixes all possible permutations of 1, 2,... n. But the reasoning of Art. 8 shews that each term in the new determinant has the same sign as the corresponding one in the original determinant. Thus a determinant remains unchanged in value when its rows and columns are interchanged. Alternate Numbers. 12. The magnitudes with which we deal in ordinary or arithmetical algebra are subject, as regards their addition and multiplication, to the following principal laws: (i) The associative law, which states that (a + b)+ c = a + (b +c) = a+ b + c, and that ab. c = a. bc = abc. (ii) The commutative law, which states that a + b =b + a, ab = ba. 10-13] ALTERNATE NUMBERS 17 (iii) The distributive law, which states that (b + c) a = ba + ca, a (b + c) = ab + ac. The researches of modern algebraists have led them to consider quantities for which one or more of these laws ceases to hold, or for which one or more of these laws assumes a different. form. Numbers, whether real or ideal, which follow the laws of arithmetical algebra will be called scalar quantities. We shall find it usefill to consider a class of numbers which have received the name of alternate numbers. These are determined by means of a system of independent units given in sets like the co-ordinates of a point in space; such a set will be denoted by e1, e2,... e-. A number such as A = aale + a2e +... + anen, formed by adding the units together, each multiplied by a scalar, will be called an alternate number of the nth order. In combination with scalar quantities and with units of other sets these units follow the laws of ordinary algebra. In combination with each other the units of a system follow the associative law and the commutative law as regards addition, but for multiplication we have the new equation ere =- ee.............................( ), when r, s are unequal; and er = 0................................... (2) for all values of r. 13. If A = ael+ a2e2 +... + a.e, B = b1e + b2e2 +... + bnen be two alternate numbers of the nth order, we define their product as follows: AB = arerEbes = a,.er. b es r, s = Ea,.bseres. ~~S. D,~rs S. D. 2 18 THEORY OF DETERMINANTS [CHAP. II. Hence, by equations (1) and (2) of Art. 12, AB (ab2 - a2b) ele2 + (ab3 - a3 b) e1e3 +... + (an-l b - anbn-_) en-_en Thus clearly AB =-BA and A2 = 0, proving that alternate numbers have the same commutative law of multiplication as the units. 14. If k be any scalar (A + kB) B = AB + kB2 = AB, so that the product of two alternate numbers is not altered if one be increased by a multiple of the other. If we have a product of more than two numbers ABC...... L, it follows that for one of them, say C, we can write C + kAA + kB +... + k.L,:and the product will still remain unaltered. Alternate numbers belong to that class of algebraical magnitudes for which multiplication is a determinate, but division an indeterminate process. In fact AB - =A + kB, where k is an arbitrary scalar. The continued product ele2... en of all the units of a set will in future be assumed to be unity. An explanation of this assumption will be given later on. 15. If, now, we take a square array of elements such as that in Art. 8, we can form a system of n alternate numbers of the nth order by taking the elements of each row to form the coefficients of the units in the numbers. Let P be the product of all these numbers, so that P = (aniel + a12e2 +... + ainen) (a,2 e + a,22e +... + ea2ne).. (a, el + an2e2 +...+ a e,). On multiplying out the factors on the right, P= - alpa2q... ansepeq... e8. 13-16] ALTERNATE NUMBERS 19 Since epeq... e, = 0 if any two units are alike, it follows that in every term on the right which does not vanish p, q... s is a permutation of 1, 2... n. It follows at once from the law of multiplication (equation (1), Art. 12) that epeq. e, =( e... e- e..en, where v is the number of inversions in the series epeq... es. Thus P = ele2... eG ( (- 1)v alpa2q... an, but the term under the summation sign is a term of the determinant of the system of elements, with its proper sign. Thus P- =a,,n I ele2... en -ann * Hence the determinant of a system of n2 elements is expressed as a product of n alternate numbers linear in these elements. From this it immediately follows that if all the elements of a row are multiplied by the same number the determinant is multiplied by that number, and if all the elements of a row vanish the determinant vanishes. In future we shall write for a determinant of the nth order whichever of the forms (IAr = a\e + arae2 +.. + a ) is most convenient. (Ar = arle, + ae.22 +... + arnen) is most convenient. 16. If the determinant is so constituted that the different factors of which it is composed do not contain all the units, its evaluation is frequently effected with ease. For example, the determinant an,, O, 0...... 0 a21, a22, 0...... 0 a31, a32, a33...... 0........................... amn, a12), an3 *.*-.. ann in which all the elements above the leading diagonal vanish reduces to the product a1a22... an. 2-2 20 THEORY OF DETERMINANTS [CHAP. II. For it is equal to the product of the alternate numbers ariel' a2iel + a22 e2 a^3 el + a32e2 + a33e3.......*.......... ** amel + an2e2 + a,,3e3 +... + annen,. Since the first number contains el, and el only, all terms in the product of the remaining factors which contain el disappear when multiplied by this factor, so that as far as we are concerned we may suppose a21, a,... a,, to vanish. The second number reduces to a22e2, and the product of the first two to anlela22e2. We may shew in like manner that a32, a42... may vanish, and so on. Finally the product reduces to a11 e a22e2... aennn = all,.. a.nn By an interchange of rows and columns it follows that the determinant for which all the elements below the leading diagonal vanish also reduces to its leading term. 17. As another example let us consider the determinant 0, cos (a, + a2), cos (ai + a3)...... D cos (a2 + ai), 0, COS (a2 - a3).... cos (a3 + ai), cos (3 + a2), 0.........oo.......,oo...................o o oo..oo o............ of order n: the element in the rth row and sth coluinn is cos (a,. + a,) unless r s, when it vanishes. Substitute for the cosines their exponential values and write e - a. Then D is the product of such factors as = a+ - (ala + al 2 el where E = ace, + a2e2 +... + ne,,, F el +... +. a, a2 an 16-17] EXAMPLES 21 Thus if asE + = As, we see that (- 2)n D = -I (2e, cos 2a, - A8). Now observe that since the quantities As depend only on the two alternate numbers E and F, the product of more than two of them must vanish. Hence expanding ee,...An (-2)nD=2ncos 2aicos 2a2... cos 2an- 2n cos2a... cos 2an 2 a 2e COS 2eA1 + 2n cos 2al... cos 2an eles 2.. e-2 An-1An 4 cos 2ai1 cos 2an, Now ee2... en- An = e... en-_ (aE + n ) =4sin(a1Can)n Thus 1 -( D - D sinn (a8.- a,) 1)-an- an,= (- 4 sin (a,, - a.) Thus - -- --- =)~ D = 1 - - - SIn2(a,. -a —,) cos 2al cos 2a2... cos 2an 1 cos 2ar coS 2a,' (- l)n- D sin(12 (a, - a,) cos 2a... cos 2ad cos 2a, cos 2as' where (r, s) are all duads derived from 1, 2... n. CHAPTER III. GENERAL PROPERTIES OF DETERMINANTS. 1. IF two columns or rows of a determinant be interchanged the resulting determinant is equal in value to the original, but of opposite sign. Let D = HI (ae+... + a... + astet +... + a,.,e) = IA,.; then, if D' is the determinant got by interchanging the sth and tth columns, D' = II (arle +... + artet +... + m.en); but since in addition we follow the ordinary commutative law, D' is got from D by interchanging e, and et in the product on the right. This leaves the scalar factor unaltered but changes the sign of the product of the units, thus D' =-D. Interchanging two rows of a determinant, say the rth and sth, is the same as interchanging the two factors A,. and As on the right: this is equivalent to an odd number of inversions, and hence by the rule of multiplication changes the sign of the product. This second argument, in fact, proves both parts of the proposition, since D is unaltered by changing rows into columns (II. 11). 2. If two rows or columns of a determinant be identical the determinant vanishes. For by the interchange of the two columns in question the determinant changes sign, but both columns being alike the determinant remains the same, thus D=-D or D=0. 1-4] GENERAL PROPERTIES OF DETERMINANTS 23 3. If each element of the rth row consist of the sum of two or more numbers the determinant splits up into the sum of two or more determinants having for elements of the rth row the separate terms of the elements of the rth row of the given determinant. For if D = TAs, and Ar = (ar, + b,.) el + (a2, + br2) e2 +... + (arn + brn) en = (al.e, +... + a,.nen) + (brie, +... + bnen) = A'r + Br; since A... A... A = A... (A', + Br)... An = A... A'r... An + A... Br... A, we have D = -D1 + D2, where D, and D, are determinants having for elements of the rth row in the sth place a,s and b,. respectively. Repeated applications of this reasoning shew that if the elements of the rth row consist each of the sum of p elements, then the original determinant can be resolved into the sum of p determinants having for their rth rows the terms of the elements of the rth row of the given determinant. The same theorem would apply if the elements of a column consisted of the sum of elements. In fact whenever a theorem applies to rows it applies equally to columns, as these can be interchanged (II. 11). In future, when a theorem is stated with regard either to rows or to columns, it is to be understood as applying also to the other. 4. Thé value of a determinant is not altered if we add to the elements of any row the corresponding elements of another row, each multiplied by the same constant factor. For if we add to the elements of the rth row those of the sth row, each multiplied by p, the resulting determinant is A... (Ar+pAs)...As...AnAi...As.A+pA..AA+A...As...As...An = Ai... A.... As...An, the other product vanishing, since it contains two identical factors. For brevity the operation of adding corresponding elements of two rows is usually spoken of as adding the rows. 24 THEORY OF DETERMINANTS [CHAP. III. 5. The theorem of the last article is of great importance in the reduction of determinants. The following are examples of its application: (i) If corresponding elements of two rows of a determinant have a constant ratio the determinant vanishes. For we have only to multiply the elements of one row by a proper factor and subtract them from the elements of the other when all the elements in that row will vanish, and consequently the determinant vanishes. Of a similar nature are the two following theorems, which may be proved without difficulty: (ii) If the ratio of the differences of corresponding elements in the pth and qth rows to the difference of corresponding elements in the rth and sth rows be constant, then the determinant vanishes. (iii) If from the corresponding elements of 1 + 1 rows we form the Ith differences and from the corresponding elements of m + 1 rows the rth differences (the second set of rows being at least partially different from the first set); then, if the ratio of corresponding differences is constant, the determinant vanishes. (iv) Let D= U1, v1... t u62, V2... t2 Un, Vn... tn Subtract each row from the one which follows it, beginning with the last but one. Then, if zui = ui+i - Ui, we have D= u1, vi... t1 h~i, Anv... Ati Au2, AV2... At2, o................ o..,. AUn-, AVyn-.1... AtnRepeat the same operation, stopping short at the second row. 5-6] GENERAL PROPERTIES OF DETERMINANTS 25 Then, if A2ui = A'u+i - Au, D= ul, vl... tl Au,, Av1... At, A2u_, A2v... A2t t............................ A2Un-2, A2Vn-2... 2tn-2 Proceed in this way, leaving out a row each time, and we see that D) = lu, V1... tl Au,, AV1... At, A2u1, A2v... A2t.............................. An-lul An-lVl... An-it where generally: Aru = -Ar-ui+l - Ar-lhi. Suppose now that u, is a polynomial in x of degree 0, v, one of degree 1, and so on, then all the elements below the leading diagonal of D vanish, and D = U1. Avi. A2wl... /A-ltl. For example, if m (m- 1)... (m-p 1) ~mp=- --- ^ _ -—,mo=l, MI1.2l1d.d2... dr mO, m!... m = 1. d. d2... d( (m + d)o, (m + d)l... (m + d)r = dr(r+l)/2...........................,,............ (m + rd)o, (m + rd)... (in + rd)r For here At (m + td) = dt. 6. In a determinant of the form 0, 1, 1, 1... 1 a, a(2, a2?3... 1 a1, a22, a23. 1, a31, a32, a33. every element of which ars is a type can be replaced by Ars = ars + hr + ks, where hr and ks are arbitrary quantities, without altering the value of the determinant. 26 THEORY OF DETERMINANTS [CHAP. III. For multiply the first row by hr and add it to the rth row, then in this row the first element is still 1, while in place of a.8 we have ars + hr. Now multiply the first column by k8 and add it to the sth column; the element in the first row is still unchanged, while the element under discussion has become ars + h,. + ks. These transformations have left the value of the determinant unaltered. 7. We are now in a position to solve the system of linear equations al x, + al2x.2 +... + a,, xn = U1, a21xi -+ a22x2 +... + aCm n = U2,........................ e..... e amnl1 + an2-2 + * * * + annXn == Un. Take alternate units el, e2... e,; multiply the first equation by el, the second by e2,... the last by en, and add the results. We thus get A4xl + A x, +... AnOXn = U, where A, = as,,e,, and U = ues. Multiplying by A2A... An, we obtain 1 ann = ul, a12... acn a2, a22... a2n un, a.2.. nn and in general x' is obtained by substituting in the determinant lanl for the elements of the sth column the quantities ul, u,... un, and dividing the resulting determinant by lanl. A system of (n + 1) homogeneous equations in n variables may be treated in the same way. We shall return to the subject of linear equations later on. /r~. t. 8. If p rows of a determinant whose elements are rational integral functions of x become identical when x= a, then the determinant is divisible by (x - a)-l1. For subtract any one of these rows from each of the remaining p - 1 rows; the determinant remains unchanged, but now when x = a all the elements of these p- 1 new rows vanish, hence each element divides by x - a, and 6-8] GENERAL PROPERTIES OF DETERMINANTS 27 thus dividing each of the p - 1 rows by this factor we see that the determinant divides by (x- a)p-1. If when x = a the rows are not equal, but only proportional, the theorem is still true. Ex. The value of the determinant x, a......a (n rows) a, x......,...o.o...... a, a...... x is [x + (n - 1) al (X - a)n-. For if x= a the n rows all become identical, thus the determinant divides by (x- a)"-l. Adding all the rows to the first, each element in that row becomes x + (n - 1) a, this is therefore a factor in the determinant. Thus the determinant divides by { + (n - ) } (x - a)-1. This is of the sane degree as the determinant, and as the coefficient of xn in the determinant and in the product is unity the determinant rnust be equal to the product. CHAPTER IV. ON THE MINORS AND ON THE EXPANSION OF A DETERMINANT. 1. IF from the n rows of the array all 1 a12... aln a21, a22.. a2n anl, an2... ann we select any p rows, and then from the new array which these form select p columns, these when written in the form of a determinant constitute a minor of the given system. Such a minor is said to be of the pth order. Since we can select p rows from n in n (n-1)..; (n-p + 1) 1.2...p p ways, and p columns from n columns in a like number of ways, it follows that the given system of order n ias (np)2 minors of order p. 2. If out of the n -p rows which remain after the above p have been selected we take those n-p columns whose column suffixes are different from those selected in the minor of order p, we have another determinant of order n-p said to be complementary to that of order p. 1-4] ON THE MINORS AND EXPANSION OF A DETERMINANT 29 For example, in the determinant all, al2, a13, a14, a,5 a21, a22, a23.... a25 a51, a62......... aC55 a33, a34, a35 a and a437, a4, a4s a2,, a2, a,53, a54, a5s are complementary minors. 3. If p = 1, i.e. if we take a single element, the complementary minor is à determinant of order n- 1, which is called the complement of the element. This complement is obtained from the original determinant by omitting the row and column in which the selected element stands. For example, the complement of the element ars, which we denote by Ar, is ai,.... al,s-l, al,s+,... ail,,, ar-1,1... C —, s-1,, r-1,S+l *.*. * r-l,, a,,, ar+l, a ~ +l ar+, s- +1 a+'l, n...................................... an,, * *, s-, an, s+l ** * an, n This is sometimes spoken of as a first minor of the given determinant. In like manner the determinant formed by omitting p rows and p columns would be called a pth minor; it is to be observed that a pth minor is a determinant of order n -p. 4. We may extend the meaning of complementary minors as follows: From the array in Art. 1 select p rows and p columns, then from those that remain q rows and q columns, from those that remain r rows and r columns, and so on. With the elements in these selected rows and columns form determinants; these will form a complementary system of minors if p + q + r...n. The number of ways in which we can form such a system is p! q!!.. 30 THEORY OF DETERMINANTS [CHAP. IV. It is of course permissible that one or more of the numbers p, q, r... should be unity; the corresponding minor is then a single element. For the determinant an... a16 a6G... a66 the minors ad24, a25. a12, ai3, a16 a4, a5; a42, a43, a46; a51 a34, a35 a62, a63, a66 form such a complementary system, and there are 3600 systems of this type. 5. We have hitherto only considered the product of a set of alternate numbers equal in number to the number of units. Let us now consider the product (anle, + a12e2 +... + aine)... (amleL + am2e2 +... + amn,e,); this is equal to iapa2q... amrepeq... e1., where p, q... r consist of all combinations n at a time from 1, 2... n, repetitions being allowed. First, if m > n, we must have repetitions in every term of the sum, and hence [II. 12, equation (2)] the whole vanishes. If r, = n, we have the case of II. 15, and the sum is the determinant | a.. But if m < n, the sum is formed by taking for p, q... r all rm-ads from 1, 2... n and permuting the elements of each m-ad in all possible ways. Namely, the term a1p a2q... amr. eeq... er. is got by taking apep from the first factor of the product, aseq from the second..., and arrer from the last factor. But we should still get the product of the units epeq... er, though in a different order, if we take the pth term of some other factor than the first, the qth 4-5] ON THE MINORS AND EXPANSION OF A DETERMINANT 31 of some other than the second, and so on. The term of the product which multiplies epeq... er is thus got from ap a2q... amnr by permuting p, q... r in all possible ways, and giving to each term the sign corresponding to the number of inversions in its second suffixes, p, q... r being considered the original order. The sum of these products is alp, atlq... air a2p, a2q... a2r amp, amq.* amr Hence the product of the m factors is equal to alp, alq... al epe..................(1). a2p, a2q... a2r amp, amq... am In like manner, if we take the remaining factors necessary to form the determinant I a,,n, we have (am+l, el +... + a,,+,n en)... (an,1 el +... + an,n e) =2 am+l,u, am+i,v... am,,w e, ev... e,............ ), am+2, u, am+2,v... am+2, w an, u, ) an, v.. an, iV where u, v... w is a combination of n - m numbers selected from 1, 2... n. Now multiply the equation (2) by the equation (1) and we obtain ann l = {(-1) a1., alq... air a+l,... a*+l,w }; amp, amq ~. amn a,,,, u... an,, 1 where from the nature of the alternate numbers e it follows that the two determinant factors under the summation sign are complementary minors, and v is the number of inversions in epeq... ereev... e or in p, q... r, u, v... w. 32 THEORY OF DETERMINANTS [CHAP. IV. This theorem, usually called Laplace's theorem, gives the expansion of a determinant in the form of a sum of products of complementary minors. It is assumed in the above that the complementary minors are formed from the first m and last n - m rows. Since by a suitable change of the order of the rows and sign of the determinant any m rows can be brought into the first m places, this is no real restriction. 6. For example, we have ai, aa,, 3,4 = (12) (34) + (23) (14) +(31) (24) bl, b2 b3, b4 + (34) (12) + (14) (23) + (24)(31), C1, C2, c3, C4 dl, d2, d3, d4 where for brevity (12) (34)= b, b2 d3 d4 &c. In like manner a, a2, a3, a4, a, = (123)(45)+(142)(35)+(134)(25)+(243)(15) bl, b2, b3, b4, b5 +(125) (34) +(315) (24) + (23) (14) ci, C2....... +(145) (23)+ (425) (13) d-, d2. +(345) (12), el, e............... where (123) (45)= a, a2, a d4, d5 &c bl, b2, b33 e4, e5 c 1 2, C3 7. If when the determinant is divided into two sets of m and n-m rows there are n - m columns of zeros in the set of m rows, the determinant reduces to the product of the minor of the remaining m columns and its complementary minor. This is clear, for with the exception of this single minor of order m all the others vanish because they contain at least one column of zero elements. 5-8] ON THE MINORS AND EXPANSION OF A DETERMINANT 33 If the set of m rows contains more than n - m columns of zeros the determinant vanishes. Thus, for example: a1, na, 0, O == ac, a2 C3, C4 bl, b2 0, O bi, b2 d, d4 C1, C2, C3, C4 dl, d2, d3, d4 while ai, a2, 0, O, =0. bl, b,, O, 0, 0 C1, C2, 0, 0, 0 dl, d2, d3, C4, d5 el, e2, e3, e4, e5 8. In Art. 5 we resolved a determinant into the sum of products of pairs of complementary minors. We can however resolve it into a sum of products of as many complementary minors as we please. For we can divide up the n factors whose product is a,,, as follows: Take the first u, the second v..., the last w. The product of the first u factors would be of the form 2 _ alp, aiq... ar epeq... er, a2p, a2q... a2r aup, auq... aCur or Duep eq... e., p, q... r being u numbers taken from 1, 2... n without repetition and Du a minor of order u from the first u rows. In like manner the product of the next v factors would be E Dvefe,... eh, Dv being a minor of order v chosen from the v rows. Lastly, the product of the w factors would be EDwe.es... et, with a similar meaning for the quantities involved. s. D. 3 34 THEORY OF DETERMINANTS [CHAP. IV. Now form the product of all the factors, taking care to keep then in their proper order; then lann = DtDv... DW, where Du, Dv,... Dw form a system of complementary minors of the determinant lan,. The sign of the term is determined from the number of inversions in p, q... r, f, g...h, r, s...t. 9. If in Art. 5 we restrict the first product to the single factor arie, + a,2e2 +... + a.e.....................(l), the second product becomes ArEl + A,.E2 +.... + A,nEn............... (2), where Ar, is the complement of acr (Art. 3) and Es = ele2... ees-es+... en. For we get a term of the product by leaving out each unit such as es in turn, i.e. by forming a determinant with the remaining n - 1 columns; and since we previously omitted the rth row of the given determinant, this determinant is Ars. Now multiply the n- 1 factors which form (2) by the remaining factor (1); we obtain (- 1 )r-l | ann = arA. - aA,2 +... + (- 1)8-1 ar.sA.s +.... For esEs = es. e1... esles+... e =(-1)S-~e...e.=(-_)l-l, eEt = O if s is not equal to t. The factor (-1)'-1 on the left is accounted for in the same way. Thus l an I=; (-)a+SatsA,,s 8-10] ON THE MINORS AND EXPANSION OF A DETERMINANT 35 For example, a, a2, a3, a4 = c a2, a3, a4 - c2 al, a3, a4 bl, b2, b3, b4, 4 b, b 4 bl, b3, b4 C1, C2, c3, c4 d2, d3, d4 d1, d3, d4 dl, d2, d3, d4 + C3 al, a2, a4 - C4 ai, a2, a3 bl, b2, b4 bl, b2, b3 d d d d, d, d,, d3 10. In the final equation of Art. 9 As, is got from la,,l by erasing the rth row and sth coluînn and writing the remainder as a determinant. It is however more symmetrical, and sometimes convenient, to give to A,. a different form obtained by a series of cyclical permutations of rows and columns. In Ar remove the first row by a series of interchanges to the last place, then move what is now the first row to the last place, and so on, until we arrive at what was the (r - l)th row, which we remove to the last place. This introduces (r - 1) (n - 2) changes of sign. Now remove the first column to the last place, and so on, s - 1 times, necessitating (s - 1) (n - 2) changes of sign. In all we have introduced (r - 1)(n - 2) + (s - 1)(n-2), or (r + s) n changes of sign (an even number of changes being neglected). So that, if the new determinant is called A'rs, we have A, = (- l)_ (r+s) A,, -A,~s — 1 ~) Ars, and lan, = (- 1)(+) (r+s) arA ',, where a,.+l,s+l, ar,+ls+2,,,. a+l,,nr+l,..l. a 'r+l,s-.................................,,,........... aA', s+=, a,+2 * l,) 1 an, an, 1 * an,ar-1...,., ar-1, s+2... ar-., nar-,... ar.-, s-1 3-2 36 THEORY OF DETERMINANTS [CHAP. IV. For example, a1, a2, a -=bi c, c.3 +b3 c3, c 2 3 + 3 c1, C2 bl, b2, b3 2, a33 a3 a 3, a2 Ci, C.2, C3 In future we shall always write annL = a arsArs, s and suppose that Ars has its proper sign. 11. We may arrange the complements of the elements of a determinant in another square array, and then the two arrays a...... ai Ail....... A n......(.................(.................. (2), a...... a j A...... A nn are said to be reciprocal. (a; ' If now a sum be formed by multiplying each element of a row of (1) by the corresponding element of a row of (2), and adding these products together, the sum is equal to the original determinant or zero, according as the two rows have the same suffix or not. Narnely, ao.1Asi + a,.2A2 +.. + ar, As-n= annl or 0, according as r is or is not equal to s. For if r is equal to s the sum on the left is the expansion of the determinant according to the elements of the rth row, but if r is not equal to s the sum on the left is what the expansion of the determinant would be, if its rth and sth rows were identical, but if the elements of two rows are identical the determinant vanishes. In like manner, if we multiply the elements of a column of (1) by the corresponding elements of a column of (2), we get airA ls + a2rA +s.. + acn A is, and this sum is equal to U(n or 0, according as r is or is not equal' tos. *1 Kronecker has introduced the symbol k-s to indicate 1 or 0 according as;he integers r, s are equal or unequal. With this. notation, we have 2atrAts Srs tnll,),^ h M (''r,,,/r 10-13] ON THE MINORS AND EXPANSION OF A DETERMINANT 37 12. If all the elements of a row vanish the determinant vanishes, as we see at once by expanding the determinant according to the elements of that row. If all but one vanish the determinant reduces to the product of that element and its complement; viz. if all the elements of the rth row vanish except ars, then the determinant reduces to asA,,. Thus for example, a11, ai, -. ai, -= a a22. a 0,, 0, a2... a2n........... O, a,2... a3h an2.. ann................ 0, an2... an an a, ai3... an = a I a22, a23... 0, a22 a23...a a2 0, a33... a3 0, 0, a3s...a3n..... 0................., O..... an 0, 0, O..... = aan 22 a33. a3n 0... ann = alla22a33.. ann 13. The theorem of the preceding article is of use in evaluating a determinant by reducing it to one of lower order. If the determinant is not of the required form to begin with, it can sometimes be reduced to it. We may exemplify this by finding the value of the determinant Dr- 0, a, a...a b, O, a...a b, b, O...a............ b, b, b...0 (r), the suffixes denoting the order of the determinant. The elements of the leading diagonal are zero, those to the right of it ail equal tO a, and those to the left all equal to b. 38 THEORY OF DETERMINANTS [CHAP. IV. If we subtract each row from the one which follows it, beginning with the last but one, Dr= 0, a, a, a......a b, - a, O, 0......0 O, b, -a, 0......0 *.....o..................... 0, 0, 0, 0...-a (r). The first column contains only one element, hence D = - b a, a, a, a... b,-a, O, O... O, b, -a, O... 0, O, b, -a.......................... Regard the elements in the first row as a+ 0, O+ca, 0 +a... then we can resolve the determinant into the sum of two: Dr=-b a,, 0, 0... -b, a, a,,a... b,-a, O, 0 b,-a,, 0... O, b, -, 0... O, b, -a, O... 0, 0, b, - a... 0, 0, b, - a............................ (r-1).......................... (r- ) In the first of these two determinants all the elements above the leading diagonal vanish, hence its value is (-l)r- ar-l. The second determinant is of the same form as that to which we first reduced D,, hence Dr = - bDr_ + b (-a)r1. This is an equation of differences with constant coefficients for Dr, and its solution is =(- 1) - ab (a - a n a a -bthr 14. In Art. 11 we saw how under certain circumstances the order 13-15] ON THE MINORS AND EXPANSION OF A DETERMINANT 39 of a determinant might be reduced. Conversely we are enabled to increase the order of a determinant without altering its value, namely, by bordering it with a new row and column in one of which all the elements vanish except that cominon to the other. Thus ann,= 1, 0, O,... X, all, a1, a3,... y, a21, a22, a23 ' = (_ l)n o, 0, O... 0, 1 ali, a 12, 13... aln) X a21, a22, a23. a2n y................ where the quantities x, y... are any whatever. By adding on to these a new row and column we can raise the order of the determinant to n + 2 and so on. 15. In the determinant D=la l, if we suppose only the element ars to vary, since on expanding according to the elements of the rth row D = arlArl + arC2A2 +... + arsArs +. the only variable term on the right is the product a,,sArs, we see at once that aa,, aD If among the elements of Ars only afg is variable, we see that aAr,_ 2D aafg aafga.rs a2D Thus aa afgar is the sum of all terms in D which contain the product afgars. The differential coefficient a2D Pa7ar 40 THEORY OF DETERMINANTS [CHAP. IV. is the determinant obtained by erasing in D the rth andfth rows and the sth and gth columns, and is therefore complementary to ars, a,, af, afg In like manner it is plain that an-mD amD 7 --- and Daf,rags... and puOaqv... are complementary determinants if f, g...p, q... r, s... u, v... are each of them permutations of 1, 2... n, i.e. if the product afrags... apa qv... is a term of the determinant D. 16. If all the elements of a determinant are functiiLf variable t we see that dD ) D dars dt = a. ' dt (r, s, 2.. t) If we denote differential coefficients with respect to t by accents we have D' = A,1 a'r + SZAa'. +.. a '1, a22... a2 C a2, 22, *. a2n So that D' ish tLof n detersminiboaflag indb. substit2 for the elem s f each colu m - n-ee Si coefficients withrespe2tl An interesting example of this is to consider the differential coefficient of D=- u, u,,... u(1) V, V', V",.. V(n-1) w,, ",... w (n —)................ents denoting differ........ential coefficients with respect to t accents denoting differential coefficients with respect to t. 15-17] ON THE MINORS AND EXPANSION OF A DETERMINANT 41 Each of the first n -1 determinants obtained by the preceding rule vanishes because it has two columns alike, the last alone does not vanish, so that dD _ t, U'... U(n-2), (n) dt v, V'. v(n-2) (n) Wq, W'... W(-2) W (n) As another example take the determinant Dn= 1, 1... 1 tl, t2... tn tl2, t22... tn tln-i, t2n-... t -l1 oD, Then at is got from D, by substituting for the elements of the atr rth column 0, 1, 2tr, 3tr... (n - 1)t-2. Hence _a-lDn O, O... O, 1 a3tlt2... atn-x 1,...t 2tl, i2t2... 2tn-1 tn2 (n-1) tl-2, (n -1) t;-2... ( - 1):tn-, tn'-1 -(- l)n- (n-l)! lD_n. 17. We may use the theorems of Art. 11 of the present chapter to prove those of Arts. 3 and 4 of Chap. IiI. If each element of a row of a determinant is the sum of p terms, the determinant is equal to the sum of p determinants having for their elements the separate terms of the sum in question. For if ar= ps + q +... + t, then lann\ = 2asrs = EpA,Ar + qsA-rs +.. + ts Ars =P+Q+... + T, 42 THEORY OF DETERMINANTS [CHAP. IV. where P is the determinant obtained from the given one by writing p p2,... pn for the elements of the rth row and Q... T have similar meanings. The value of a determinant is not altered by adding to the elements of any row those of another row multiplied by a constant factor. For if to the elements of the rth row we add those of the tth row, each multiplied by p, the resulting determinant is equal to ( (ars + pats) Ars = arArs r + peats Ars s = 1annl the last sum vanishing by Art. 11. 18. If each elment of a deteinant consists of the sum of p terms, we could by continued application of the first theorem in Art. 17 reduce this determinant to a sum of determinants whose elements are all single terms. But a formula of expansion has been given by Albeiani which presents the result in a more suitable form for applications. Let ars = arsI + ars2 +... + arsp, so that each element in the determinant is the sum of p terms. Then each column of the determinant when written at full length would consist of p partial columns whose suffixes are the third suffixes of the above elements. With these partial columns we can form p determinants, taking all the partial columns with the third suffix 1 to form the first, those with the third suffix 2 to form the second, and so on. We shall denote these determinants by DDl, _D(nl)... Dp(n) so that Du(n) = ali, al2u.. ainu a21U, a22u... a2nU..................., anu, an2u.. annu The first two suffixes tell us the row and column in which the element stands, the third the determinant to which it belongs. The original determinant is denoted by D~. The index in brackets tells us the order of the determinant. 17-20] ON THE MINORS AND EXPANSION OF A DETERMINANT 43 19. We shall find it necessary to employ the term complementary minors in the following sense. From the elements of D1(n), form a minor Di(") of order a by selecting a rows and columns. Then in D2,(> select 3 rows and columns, whose suffixes are different from those selected to form D9l(), these form a determinant D,2(), and so on until we take 7r rows and columns from Dp(n), to form a determinant Dp), none of which have the same suffix as any of the preceding. Then if a+ +y+7 4... +7r=.....................(1), D(), D2(), D3()... Dp shall be called a series of comment mir. Any one or more of the numbers a, /... 7r may be unity or zero. 20. We shall now prove that D() = SD1(a)D2 )...D () where the meanings of the summation signs will be explained presently. For we have D() = II (arle + ar2e2 +... + aj.enn), and if Csas = arisel + ar2e2 +... + as,,e,, D ) = Il (Url + utr2 +... + )........... (2), the product containing n factors. We shall obtain a term of the product on the right if we take a factors such as un, 3 factors such as UZ2,... 7r factors such as ulp, provided the equation (1) is satisfied. But from the definition of a determinant this product of factors is equal to a determinant of order n the first a of whose rows come from D,(n), the next 3 from D2(),... the last wr from Dp(n). Expand this determinant in the sum of products of complementary minors of order a, /... 7r selecting the rows of the minors from the first ca, the next /,... the last 7r, its value is then (Art. 8) ED1 (~a)D2().).D(), with the notation of Art. 19, and the summation sign means that we are to take all the possible complementary minors. 44 THEORY OF DETERMINANTS [CHAP. IV. This is only a single term in the expansion of the product (2); the whole product is obtained by summing this for all values of a,... 7r which satisfy the equation (1). Thus D( = S- D () D)................. (3). 21. The number of terms in the sum E is n! a/3!... 7r!' Let us compare the expansion (3) with the expansion of the multinomial (D~ + D +... + D)n. The general term is CD1aD2.D (4), 1 PD,uD...D........................ (4), where a, 3... 7r satisfy (1) and n! a!/!... 7r! Comparing (3) and (4) we see that in expanding the determinant we replace C by ', and a, 3... wr are no longer exponents, but merely indicate the orders of the determinants D1a), D2(a,; etc. Hence we may writ svmbolicaly for the expansion of our determinant where in every ter of the ltiomial expansion we replace where in every term of the multinomial expansion we replace the coefficient by a summation sign, the number of terms in the sum being given by the multinomial coefficient and the exponents a,/3... vr now indicating the orders of the complementary minors. Thus finally we have the symbolical equation D() =(D1 + DJ +... + Dp). 22. Let us make use of this theorem to expand the determinant D = an + z4, a2, a,... a. a21, 22 a ~ - 2 a23 ** a2n a31 > a32 CC3 4+ Z3... a3n anl, an2, an3... aCnn + Zn according to products of the quantities z, z2... Zn. 20-23] ON THE MINORS AND EXPANSION OF A DETERMINANT 45 Here we must write D( = a1... an D(n)= z1,... O............ O, z,... 0 a,,,. a nn............... O, 0... Zn Then by the above theorem D = (D + D2)" = Dl(n) + EDl(n-1)D2(1) + ED1(n-2)D2(2' +... + D2('1). Now clearly all minors of D2~> vanish except those whose leading diagonal is part of the leading diagonal of Dn). Thus D2(1) = Zi, D2(2) = ZiZk,... D2) = z22... Zn. The corresponding minors D2(n-1), Dl'-2)... are got by erasing in D<(>) the ith row and column, the ith and kth rows and columns, &c. Thus D = D1(?n) + zi D1(n1-l) + ZiZkD(12 + Ez+. z +... z n. Or if we simply denote Dl(2) by D1, D = Di + Z i3k d -i + z + ZiZ2... Zn aai aaiiakk If 1 = z... = we get DD, =+Z25 + D= D +zE x -C+ Z n -+... +2z. 8aa^i a@aiiakk These results may also be obtained by using the generalised form of Taylor's theorem. 23. Any determinant can be written in the form D = 0+ all, a12... aln a2,,O + a22... a2n.......... o~~......oooo am, an2... O + ann We may now apply the theorem of Art. 22 by supposing D1 = 0, a2... aln a21, O... a2n an, an2... O and zi = ai. 46 THEORY OF DETERMINANTS [CHAP. IV. Then D = D - + aii -a iiakk + a ---. + a11a22... an, aai oaajaackk the general term being aii akk... aCr D, (n -m) where D,(n-m) is the minor obtained from D1 by suppressing the ith, kth... rth rows and columns, m in number. It is clear that D1 () is zero, for the suppression of (n- 1) corresponding rows and columns of DI leaves us with one of the zero elements in the leading diagonal. Ex. If 0, ali = (12),&c. a21, 0 we have an... a14 = ala22a33a44 +- ala.22 (34) + alla33 (24) + alna44 (23).......... + a22a (14) + a22a44 (13) + a3344 (12) a41... a44 + a 1(234) + a,2 (134) + a33 (124) + a44 (123) + (1234). As another example we may find the value of the determinant D= ci 1, a, a, a... a b, c, a, a... a b, b, C3, a... a oooo............. b, b, b, b... cn The general term in the expansion of this determinant is 'Ci Ck. Cr Dl (n-m), where ci, Ck... cr. are any nm elements of the leading diagonal. But by Art. 13 ab D(t-n) = (- _ )n-m- n-m- _ bn-m-.); whence, if f (x)= (ci - x) (c2 - x)... (cn- ), it is clear that D af(b)- bf(a) a-b 23-25] ON THE MINORS AND EXPANSION OF A DETERMINANT 47 If we write down the similar determinant of order n + 1, for which cn, = 0, after dividing both sides by ab, we get cl, a... a, 1 f(a)-f(b) b, c...a, 1 a-b b, b... cn, 1 1, 1... 1, If we now put b = a, we get a determinant expression for f'(a). 24. We have seen how to expand a determinant according to the elements of a row or column. It is frequently useful to be able to expand a determinant accdi the elements of a row and column. This is effected by means of the following theorem Ve to Cauchy, nna = ars As - a,.kais Bik which expands a determinant as an explicit function of the elements which occupy the rth row and sth columnn. A,s is the complement of ar and Bik is the complement of aik in As,, and is therefore a second minor of the original determinant. For every term which does not contain a,. must contain some other element from the rth row and some other element from the sth column, and hence contains such a product as arkais, where i and k are different from r and s respectively. The aggregate of all terms which multiply a,. is Ars; now a,.kais differs from arsaik by the interchange of the suffixes k and s, thus the aggregate of terms which multiplies arkais differs in sign only from that which multiplies a,,aik, that is to say, differs in sign only from the coefficient of aik in Ars. Hence - Bk is the coefficient in question. 25. This theorem is useful for expanding a determinant which has been bordered. For example by this theorem D= bpq, bp, bp2... biq, a1i, a,... b2q, a21, a.22 = bpq a -nn - bpkbiqAik, where Aik is the complement of aik in | an |. 48 THEORY OF DETERMINANTS [CHAP. IV. By the selection of a suitable bordering we are often able to evaluate a determinant by means of this theorem. For example, let al= x, a2, 23... an ai, x2, a3... a ai, a., x3... a.................... ai, a2, a3... n all the elements in the ith column being ai except that in the ith row, which is xi. Then by Art. 14 D= 1, O,,... 1, x1, a2, a,... 1 a, xl, s a3.. 1, a2, aX, 3.................. Multiply the first column by ai, and subtract it from the ith column; do this for each column, the value of the determinant is unaltered, and D= 1, - ai, - a, -3, 1, x1-ai, 0,, 1, 0, x2 - a2, 0 1, 0, 0 - a-,... Here the bordered determinant is xi - a O, 0... O, X2 - a2, 0... -0, a1,, O O, O, x3- 3..................................,. for which all first minors vanish except those of diagonal elements. Hence, in the theorem of this article, we must suppose i = k; if f= (x1 - a) (x2 - a2)... (x - a), f' (x)a it follows that D =f+ Ea,.f' (), a theorem due to Sardi. CHAPTER V. COMPOSITION OF ARRAYS. MULTIPLICATION OF DETERMINANTS. 1. IN dealing with rectangular arrays it is often convenient to use an abbreviated notation. The array a,,, a,,2 *. ain a21, a22... a.rn................... ami,, am2... amn with m rows and n columns is said to be of the type m x n, and may be denoted by the symbol (an). Associated with this is the array a11, a21... ami aL2, a 22...* *.* îm*....* ***** ain, a2n... a,., which is called the conjugate of (amn), and will be denoted by (amn)'. This is of type n x m, and its conjugate is (an). A square array of type n x n in which all the elements are zero except those in the leading diagonal may be denoted by [annl. Another way of writing it is (8nnan). The array (ann) is said to be deficient if m < n; redundant if m > n. When rn = n we have a square array. The conjugate of a redundant array is deficient, and vice versa. Two arrays of the same type may be combined into a sum or difference according to the rules expressed by (aS.n) ~ (bmn) = (mn ~ bmn)4 S. D. 4 50 THEORY OF DETERMINANTS [CHAP. V. If k is any constant quantity, the product of k and (amn) is defined by the formula k x (amn) = (Cn) X k = (a,,). 2. Suppose, now, that (a,,), (bp) are any two rectangular arrays of types m x n and n x p respectively. Let Cik = aibk + ai2b2k +... + ainbnk; then there are m x p quantities cik, which may be regarded as the elements of an array (c,p). We shall write (Cnp) = (am-n) (bnp). It is to be carefully observed that the order of factors on the right is essential. According to the definition (bnp) (amn) has no meaning unless m=p; and even then the meaning of the symbol is, in general, different from that of (a,,,,,) (bnp). 3. When p = m, the array (Cmp) is square; we propose to find an expression for the determinant c,,m.Taking m alternate units el, e,,... em, we have Ci = cile, + +cie2.. + cimem = ai Bi + ai2B +... + aiBn,, where Bk = bklel + bke2... + bkmem. Hence lcmml = l Ci = n (ailB + ai2B +... + anBn). There are now three cases to consider: (i) If m > n, the product last written vanishes, because in each term of the expansion at least one factor Bk is repeated. (ii) If m==n, Cnn1 = annl I1Bk =1 |nn. | bnn, (iii) If mn <n, the product on the right is the sum of such terns as al, aq, air... BpBqBr a2p, a2q, a2r amp, amq, amr.. where p, q, r... are m numbers taken from 1, 2... n (iv. 5). 1-4] ON THE MULTIPLICATION OF DETERMINANTS 51 But pBqBr.= bpl, bp2... bpm ee2.. em, bqi, bq2... bqm bri, br2... b,..,...,........... so that, finally, cmml- ap, aiq, air...*** bp, bp2... bpm a2p, a2q, a2r.. bql, bq2... bqm......................br, br... brm amp, amq, amr -n.................. where for p, q, r... we are to write all possible m-ads from the n numbers 1, 2... n. 4. The second case of Art. 3 gives us a rule for expressing the product of two determinants of the nth order as another determinant of the same order; the rule being |annl. |bnnl = \Cnnl, where cik = ailbik + ai2b2k +... + ainbk. Since the element cik is derived from the ith row of Ial and the kth column of Ibnn the product formed by this rule is said to be effected according to the rows of the first determinant and columns of the second. But since in either or both of the determinants ial, blnnl we may interchange rows and columns without affecting their value, we see that the product of two determinants can be obtained in the form of a determinant in four different ways, viz. the element Cik has one of the four forms: aiablk +..b +. + -ainbnk, ail bkl + ai2bk.2 +.. + ain bkn, alibik + a2ib2k -4-... + aibnk, ali bki + a2i bk +... + anibkn, where we multiply the elements of a row of | anl by the corresponding elements of a row or column of Ibnn\; or the elements of a column of lann, by the corresponding eleinents of a column or row of Ibnn. There are really only two essentially distinct cases: multiplying by rows, when we multiply corresponding elements of 4-2 52 THEORY OF DETERMINANTS [CHAP. V. two rows together; and multiplying by rows and columns, when we multiply the elements of a row by the corresponding elements of a column. The four forms of the product correspond to the four compositions (ann) (bnn), (ann) (bn, (an)' (bnn), (ann) (bnn)'. The product of two determinants of orders n and m (n > m) can be expressed as a determinant of order n by applying the process of Iv. 14 to increase the order of one of them until it is equal to that of the other. 5. Examples. Compounding the two systems ai, bl, cI pi, p2 a2, b2, c2 ql, q2 ri, r2 we get the theorem a1pi + biqi + clri, alp2 + b1q2 + Clr2 a2pi + b2qi + c2rl, a2p2 + b2q2 + c2r2 al, blI p q aq, ci 1 p, ri+ b1, Ci q ri a2, b2 p, q2 a2, c p2, r2 b2, c2 q2, r2 while if we compound the systems a1, a2 pi, ql, -r b1, b2 P2, q2, r'2 C1, C2 we get alpi + a2p, alq1 + a2q2, a1r, + a2r2 = O. blpl + b2p2, b1ql + b2q2, blrl + b2r2 C1pl' + C2_p2, Clql + C2q2, Clrl + c2r2 Again, the product of the two determinants ai, bl, c1 pi, ql, ri a2, b2, c2 p2, q2, r2 a3, b3, c3 3, q3, r3 4-6] ON THE MULTIPLICATION OF DETERMINANTS 53 is the determinant alpi + b1qi + clri, alp2 + b1q2 + cr.,, alp3 + b1q3 + c1r, a2pl + b2q1 + c2r1, a2p2 + b2q2 + c2r, a2p3 + b2q3 + c2r a1p2 + b3q1 + c3l1, a3p2 + b3q2 + c2r2, a3p3 + b3q3 + c3r3 while ai, bl, c1, di. pi, qli = ai, bl, c i, d pi, ql, O, a2, b2, C2, dl p,2 q2 a2, b2, c2, d2 p2, q2, O, 0 a3, b3, C3, d3 a3, b3, c, d3 O, O, 1, 0 a4, b4, C4, d4 a4, b4, 4, O, 0, O, 1 (forming the product by rows and columns) = alp + b1p2 a2lq + b1q2, cl, dl a,2p + b2p2, a2q1 + b2q2, 02, d2 a3pl + b3p2, asq1 + b3q2, c3, d3 a4p1 + b4p2, a4q1 + b4q2, C4, d4 Multiplying by rows we have a, b c, d ac + bd, - ad'+ bc' -b', a' -d', c' -b'c + a'd, b'd' + a'c' Now let a, b, c, d be the complex numbers a= x + iy b =u + iv c = p iq d =r+is and a', b', c', d' their conjugates, a = x - iy, &c. On multiplying out the three determinants we have Euler's theorem concerning the product of two numbers each the sum of four squares, viz. (x2 + y2 + q2 + V2) (p2 + q2 + r2 + S2), = (px - qy + ru - sv)2 + (py + qx + rv + su)2 + (pu + qv - rx - sy)2 + (pv - qu - ry + sx)2. 6. The square array (a.) (an)' has for its elements Cik = ailak + ai2ak2 +... + ainakn = Cki, 54 THEORY OF DETERMINANTS [CHAP. V and, if m < n, mm= ap alq, air. 2 a, a2, a2q, ar* *. or the determinant is the surn of n, squares. If then the elements aik are all real the determinant ICmm can only vanish when the determinant under the summation sign on the right vanishes for all values of p, q, r.... Thus compounding ai, bl, C1 a2, b2, C2 with its conjugate, we see that a1i2 + b12 + c12, ala2 + bib2 + cci2 ai, bi 2 b1, cl 2 ai, C 2 ala2 + bb2 + c2,ic a2 + b 2 + c2 a2, b2 b2, c a2, c2 or (ai2 + b12 + c12) (a22 + b22 + c22) - (ala2 + blba + ClC2)2 = (alb2 - a2bl)2 + (blc2 - b2c1)2 + (alc2 - a2i)2. Again ai, bl, Cl 2 a2 b + b2 +c2, aa2 + bb2 + cc, ala + blb + cc3 a2, b2, c2 = ala2 +i bb2 + cc2, a22 + b22 + c22, a2a3 + b2b3 + 2C3. a3, b3, C3 ala3 + b1b3 + cic3, a2a3 + b2b3 + c2c3, a2 + b32 + c32 7. S lvester has shewn how, by the artifice of bordering the determinants as in IV. 14, the od f two detrinants of O^_._ L^el^E-_-q- forms. ^We shall order n can be ereesented in n d istinct forms. We shall illustrate this for the case n = 3. The product of the two determinants ai, bl, c1 Pi, ql, ri a2, b2, c2 p2, q2, r2 a3, b3, C3 ps, q3, r3 is the determinant of order 3: alpi + biq, + ciri, alp2 + b1q2 + cr2, alp3 + b6q3 + cir, a2pl + b2q1 + c2r1, a2p2 + b2q2 + c2r2, a2p. + b2q3 + c2r. a3pl + b3q, + cr,, a.p2, + b3q2 + cr2, a,3p + b3q3 + c3r3 6-7] ON THE MULTIPLICATION OF DETERMINANTS 55 But if before forming their product we write the determinants in the respective forms ai, b1, CI, 0 p, ql, O, ri a2, b2, c2, 0 p2, q2, 0, r2 a3, b3, C3, 0 p3, q3, O, r3 0, 0, 1,,, 01,0 their product by rows is the determinant of order 4: - aipl + Iqi, a + b, a)p2 + bq, a., c1 a2pl + b2q1, a2p2 + b2q2, ap3 + bq3, c2 a3pl + bsq, a2p2 + b1q2, a3p3 + b3q3, C3 ri r r,,, Again writing the original determinants in the forms ai, bl, Ci, 0, O pl, 0, O, q,, ri a2, b2, c2, 0, p, (), 0, q2, r2 a3, b3, c3, O, O, p3, O, O, q3, r,, 0, 0, 1, 0, 0, 0, 0 0,0,0,0, 1 0,0, 1, 0,0 their product is now the determinant of order 5: alpi, aIp2, alp2, b1, ci a2pi, a2p2, a2p3, b2, C2 a3p1, a3p2, a3p3, b3, c3 q, q2, q3, 0, 0 rl r2, r3, 0, O while writing the determinants in the forms a1, bl, c1, 0, 0, 0 1, O, 0, O, O, 0 a2, b2, C20, 0, 0, 0 01, 0, O, O, a3, b3, C3, 0, O, 0 0 0,,, 0, 0 0, 0, 0, 1, 0, O 0, 0, 0, pqi, q, ri 0, 0, 0, O, 1, 0, 0, O, p2, q2, r 0, O, 0, 0, 0, 1 O, 0, O, p 3, r3 56 THEORY OF DETERMINANTS [CHAP. V. their product is the determinant of the sixth order ai, bl, c1, 0, 0, 0 a2, b2, c2, 0, 0, 0 a3, b3, C3, O, O, O 0, O, O, p, qi, ri 0, 0, 0, p2, q2, r2 0, 0, O, p3, q, r, This rule is interesting as giving us a complete scale whereby we may represent the product of two determinants of order n by, a determinant of any order froin n to 2n inclusive; it is also frequently useful in applications of the theory. 8. The fundamental theorem of Art. 3 regarding the determinant formed by compounding two arrays can be deduced as follows from Laplace's theorem, iv. 5. We can write the determinant cMml in the form of the determinant of order (n + m), Iv. 14, Cnl.. b... bni.*........................ Cinm... C~ nO bn*,, bnin O... 0,...0........................... O... 0, 0... 1 where Cik has the value ascribed to it in Art. 2. Now from the ith column subtract the last n coluinns multiplied respectively by ai,, ai2... then from the value of cik it follows that |c1m = Ob......n O... O, im... bnm -an1... -am, 1... 0.............................. - aL...- amn, O... 1 In the determinant on the right multiply the first m columns 7-9] ON THE MULTIPLICATION OF DETERMINANTS 57 by- 1 and then move the second m rows to the beginning, then (after m + m2 changes of sign) our determinant is equal to a,... a, 1..., O... O........................................o............ am... amm,... 1, 0... 0 0... o0...bi, bm+,,,... bn..,,o*........................................... 0... o, bim... bmm, bm+,,In... bnm ai, m+... * am, m+i O... O, 1...................................................... a,,... a n, 0... O, 0... 1 Now expand this by Laplace's theorern according to minors of the first m columns. Let us find the complement of the minor alf, a2f ag, a2g................ For this purpose we move the rows of a's having the suffixes /, g... up to the beginning; then move those columns of b's which have the suffixesf, g... into the (m + 1)t, (m + 2)nd... places. This does not alter the value or sign of the determinant, and in every place where a 1 stood before, will now again stand 1. Hence the required complement is b11, b91...O O _ b11, bg6... bf2, bg,... 0 0 bf2, bg2... 0 0..... 0 0... 0 1 O O... 01 Hence Cnmml =- alf, a2f... bf/, bgl.. al., ag... bf2, bg2 where f, g... is an m-ad from 1, 2... n. This agrees with our former result. 9. The value of any minor of order u of the determinant Icnn, the product of two determinants Iann\ and bnn,,, obtained 58 THEORY OF DETERMINANTS [CHAP. V. from (an) (bnn', say Cp=Cfp, Cfq..* Cf Cgp) Cgq... Cgs,............... Ckp, Ckq. * Cks can be expressed as the sum of products of corresponding minors of order j of the determinants an,, and I bn,I. In fact, since a < n, it follows at once from case (iii) of Art. 3 that QC=, afi, afj... afr bi, bpj... bpr ag, agj... agr bqi) bqj... bqr e....................... where i, j... r is any r/-ad from 1, 2... n. One particular case of this we shall find presently of importance; namely, when the two systerns a and b are identical, and when moreover f = p, g = q,... k = s, so that the leading diagonal of C, consists of elements from the leading diagonal of cnn. Then we see that = afi, afj.. a..f 2 agi, agj... agr a sum of n, squares. 10. The differentiiaLoe1 Ijjt oef a determinantL, elements Ci:k which is the product (effected by rows) of two determinants A, B, elements aik, bik, can be represented as the sum of products of differential coefficients of these determinants. We have AB = C..............................(1), and Cik = aibki + ai2bk2 +... + ainbkn Differentiate (1) with regard to a/p; remembering that cil, c2...ci are functions of this, we get aA 3c ac ac B = b- + b- +... + — b aaip acil aci, 2Cin Multiply this equation by aB ab - Bkp 9-10] ON THE MULTIPLICATION OF DETERMINANTS 59 and add together all the equations which can be obtained from it by writing for p the values 1, 2... n. Thus we get B. a B C Bkpblp +. + C pbnp B aE a =b i acT ac+ * But by iv. 11 all the sums on the right vanish except ZBkpbkp, which is equal to B, hence c 8A aB ac,, = ap bk ( = 1, 2... a). Sirnilarly we can prove the equations a2C 1 a2A 8 2B C i A (p, q= 1, 2... n), aCikacrs 1. 2 aaiparq abkpabsq ~ac 1 si a3A a3B aCilcaCpqacrs 1. 2. 3 aaiuapvaarw abkuabqabsw (u,v,w=, 2...n), whence the general law is obvious. CHAPTER VI. ON DETERMINANTS OF COMPOUND SYSTEMS. 1. IF the elements of a determinant are not simple quantities but themselves determinants, the determinant is called a compound determinant. Compound determinants are usually formed from the minors of one or more determinants. 2. The number of all possible minors of order m of a given determinant of order n is nm,2 (IV. 1). We can form a square array with these minors, writing in the same row all those which proceed from the same selection of rows of the given determinant, and similarly for the columns. If n,, = / and we give to the combinations of rows and columns taken to form minors the suffixes 1, 2... /L, we may denote that minor whose elements belong to the ith combination of rows and jth combination of columns, by pij, and the whole system of minors will be poll...p }....................(1). P~l... Pl,, Corresponding to each element in this array, which is a minor of the original determinant, we have a complementary minor of order n- m. We shall denote the complement of pij by qj, then these form a new array,. qil... qi.......,o...................... (2). q%. i... q. 1-4] ON DETERMINANTS OF COMPOUND SYSTEMS 61 The arrays (1) and (2) are called reciprocal arrays of the mth order. Minors of these arrays formed from the same selection of rows and columns in each are called conjugate minors. The simplest instance of two such arrays is the original system and its system of first minors, viz. all... aln An... Ain amni... ann Ani... nn. 3. If we multiply the elements of the ith row of the array (1) by the corresponding elements of the kth row of (2) the sum of the products is equal to A or zero according as i is or is not equal to k, viz. piqckl + pi2qk2 +... + pik.qk == ikA. For if i is equal to k this is nothing else than the expansion of the given determinant A according to products of minors of order m and n-m by Laplace's theorem. If i is not equal to k the sum represents the expansion of the determinant when the ith selection of rows is replaced by the cth; the rows of this determinant are not all different, hence it vanishes. The particular case ailAki + ai2Ak2 + a.. + ainAkn = 3ikA is considered in iv. 11. 4. We now proceed to investigate properties of determinants of the elements of reciprocal systems, and first we shall examine the system of the first order. Let A = ann|, D= |Ann. Forming the product of these two by rows, AD= Cnnl, where Cik = ailAki + ai2A k, +... + ain A l, and hence Cik=A or O according as i is or is not equal to k. 62 THEORY OF DETERMINANTS [CHAP. VI. Thus AD= A, O, O... =An; O, A, O... O, O, A...... D-=An-. 5. Any minor of order p in the systern Aik is equal to the complementary minor of its conjugate in A multiplied by Ap-1. Let ~- afiagk... = afi, afk... agi, agk... oo...o........ and X ~ AfiAgk... be two conjugate minors in the two systems each of order p, and let 2 + a,.asv... be the complement of + ~ afiagk.... Then, if e = 1 or - 1 according to circumstances, EA = afi, afk... afu, afv... = afiagk...aruasv... agi, agk... agu., agv........................... (1). ari, ark... aru, arv... asi, ask... asu, asv........................... We may write + aaasv... co 2 + afiagk. Now we may write 2 + AfiAgk... as the determinant of order n, Afi, Afk... AfU, AfV... Agi, Agk... Agu, Agv... O, O...1, 0... O, O...0, 1... which consists of four parts. The first square consists of the elements of +~ AfiAgk...; to the right of this is a rectangle of n- p columns and p rows containing the remaining elements of the f th, gth... rows of IA|,I. The rectangle on the left below of p columns and n-p rows consists solely of zeros, and the 4-6] ON DETERMINANTS OF COMPOUND SYSTEMS 63 square on the right of n - p rows and columns contains units in the leading diagonal and zeros elsewhere. Multiply this by the determinant A written in the form (1) above. Then (iv. 11) we have eA+ AfiAgk...= A, O... af, af... O, A... agu, agv... o.................. 0, 0... aru, arv... 0, 0... asu, asv..................... = AP ~ _+ aruasv... if we resolve the determinant on the right into products of minors of the first p and last n-p columns. Accordingly - AfiAgk... = A-1co Z + afiagk... where it is unnecessary to retain the factor e, provided that we make a suitable convention as to the signs of the determinants on each side (cf. iv. 10). From this it follows that the ratio of two minors of the saine order of the system Aik is the same as the ratio of the complementary minors of their conjugates, ~AfiAgh... co + af agh... ~AklApq... co ~ aklapq... 6. As examples of the theorem in Art. 5, we have A1... A1p = Ap-1 apfl,p+l... al,n............................. A pl... App an, p+1... ann Ap+l,p+x... Ap+l,n [= An-p-1 all... alp An,p+l... Ann ap... app The relation A ik, = -s A s co aik, ais Ark, Ars i ark, ars 64 THIEORY OF DETERMINANTS [CHAP. VI. may also be written aA aA 3A aA _2A aaik aa.s aais aak aaikaars in particular aA aA aA aA a2A aCn-i,n-i aann aan-i,n aaj, n-i aan-,in-laann If A = 0, we see that Aik, Ais =0, |Ark, Ars Aik Ais or Aorrk Ars That is to say, if the determinant vanishes, the minors of the elements of any row are proportional to the corresponding minors of the elements of any other row. 7. As an example of the use of the method of Arts. 20 and 21 of Chap. Iv., let us discuss the value of the determinant P = | aik +,lbik, aik and bik being elements of two determinants of the nth order A ( = I ak, B() a= i| bk Symbolically we can write P =(XA + 1B)n *AnBn(B+ B Now let A(<), Bl( be two determinants of order n, whose elernents are 1 aA(n) 1 B(n) = A (n) aak iki = BO b k then by Art. 4 = Aik(n) 1 1 (A (ns )n A (n) 7 and similarly Bi(n) = B. "1B(n) ~ 6-8] ON DETERMINANTS OF COMPOUND SYSTEMS 65 Or, symbolically, 1A= 1B Thus P = AnBn (XBI 4- UA)n. But (XBi + /Al1)n is the symbolical expression for a determinant of order n with binomial elements of the form X\ik + /ik. Hence, passing from symbolic to real expressions, we have the determinant equation: Xak + fibik a= k * | ik. I \ik + 4- ik. Numerous other transformations of the determinant on the left can be effected. 8. Next let us consider reciprocal arrays of order m. With the notation of Art. 2, let A = 1 |p, aà= q L, where, as before, p = n,. The product /A' is a determinant of order /L whose general element is pil qkl + pi2 q2 +... + Pg qk, which is equal to A or 0 according as i is or is not equal to k. (Art. 3.) Hence in the product determinant all the elements vanish except those in the principal diagonal. Thus A' = Ai. It follows therefore that A is a divisor of AL. Now A is a linear function of any one of its elements, hence A can only differ from a power of A by a coefficient independent of the elements of A. Among the combinations m at a time of the numbers 1, 2... n there are = (n — 1)m, which contain 1. Hence there are X elements of A, which contain an, and consequently A = xAÀ, where x does not depend on the elements of A. s. D. 5 66 THEORY OF DETERMINANTS [CHAP. VI. To determine the value of x, let ai = O except when i = k, and let aii = 1. The same will be the case with the elements P k;.'. A =1, A= 1, and.'. x=l. Thus A= A (-l)-l and '= A (n-i) for n. - (n - 1)-_1 = (n- 1)-. 9. A minor of order r of the system qi is equal to the complement of its conjugate multiplied by Ar-À. For if we multiply the determinant E + qfiqgk... by the determinant A in the same manner as we did in Art. 5 for systems of the first order, we get: A + qfiqg'k = A co E + pfipgk; and therefore, since A = Ah, q/qg = Ar- co +~ pfpg... = Ar-(n-)-l CO co + pfipgk... In like manner + p/Pk... = Ar-(n-l) co + qfiqgk.... 10. Let Ah be a minor of A, with h rows and columns. From this let us form the determinant whose elements are all the minors of order m of Ah. These last are minors of order m of A, and are consequently elements of A. Moreover, those among them which arise from the same rows or columns of A, and are hence in the same row or column of A, also arise from elements belonging to the same row or column of Ah, which is a minor of A; altogether they form a minor M of A, which has h~ rows and columns. Now by Art. 8 we have M= Ah-l)h - which gives a representation of minors of A by means of powers of minors of A. 11. If in the determinant A we select a minor Ah of order h, and form all the minors of order m in A (m > h), which contain neither all the h rows nor all the h columns of Ah, we shall form a, minor of A with n - (n - h)m_ rows and columns, which is equal to (n —L- l)n-h A{(n -1)~-l -(n —h).-h} ~_/~.A where the cmpement f in A. where AL,-h is the complement of Ah in A. 8-12] ON DETERMINANTS OF COMPOUND SYSTEMS 67 To prove this, we begin by applying the result of Art. 10, with the substitution of An-h for Ah, (n - m) for m, and A' for A. Instead of M we now have a determinant M' which is a minor of A' containing (n- h)n- = (n - h)m-h rows and columns. The value of this is (since n - h > n - m) M (n —h- 1 l)-m-l = A(n-h-1)m-h M' = A-n-h n-h Now let us consider a,, that minor of A whose elements are the complementary minors in A of the elements of M'. Since M' has for its elements all the minors of An-h which are of order (n- m) it follows that the elements of ac are all the minors of A of order m which have Ah as a minor. The order of a, is (n - h),_h, and hence by Art. 9, if a is the complement of a, in A, a- M ' A (n-1) m-i-(n-h),_-h _ (n-h-l1)..-h A(n-l) -i-(n-h7)m-h n-h *A The theorem is therefore proved, if we can shew.that a is formed as prescribed. For this purpose we must remember that ai has for elements all minors (of order m) of A which have Ah for one of their minors; to get a we have then to suppress among the combinations m at a time of the rows and columns of A all those which contain all the rows or columns of Ah; thus a has for its elements all the minors of A with m rows and columns, such that they do not contain all the h rows or columns of Ah. 12. If Ah is a minor of order h of A, and if we border it in all possible ways with m of the remaining rows and columns of A, we get the eleinents of a new determinant M, of order (n - h)m, whose value is A ^ (n-h-). A (n-h-)), -i Let us put, for the moment, n = h + k, and write A and its reciprocal in the abbreviated forms A (ahh) (bhk) | =j (A hh) (Bhc) (CMh ) V ) (Ckh) (Dkck) where! a1 = Ah. 5 2 68 THEORY OF DETERMINANTS [CHAP. VI. Consider the determinant of order km whose elements are all the minors of order (k - n) of Dkk. It follows from Art. 8 that its value is 1 Dkk j (k-1)k-m-1 = (Ak-1Ah)(k-1) m since, by Art. 5, | Dkk |= Ak-1Ah, and (k - I)k-m-1 = (k - 1)m. But any element 3ij of the determinant, as being a minor of R of order (k - n), can be expressed by Art. 5 in the form /9j = A k-f —lij, where ai is an element of Mm,. Consequently (putting km, = X) 1 3 I= MA (k-m-1)k. Equating this to the value of 1/,3, I previously obtained, and observing that (- 1) (k -l )m-(/ -l1) km =(k;-l) we find that M/7 — 1 Ah(k-1),n. A (k-1)m-l which is equivalent to the theorem stated at the beginning of this article. 13. Another way of stating the proposition is the following: If Ah is a minor of order h of A, and we form all the minors of A with m rows and columns which have it as a minor, we get the elements of a new determinant of order (n - h),_h, whose value is Ah(n-h-l)m-h. A L(n-h-l1)m-hIn the particular case of m = h + 1, the theorem is M A = A -h-. A, and the elements of M1 are determinants obtained from Ah by adding one extra row and one extra columnn: or, which is the same thing, they are those minors of A which have Ah, for a first minor. The results of this article are due to Sylvester. 14. Another modification of the theorem of Art. 12 can be obtained as follows: Let us return to the determinants A, A' of Art. 8, and form a determinant M' with the minors of An-h of order n-m; this is a minor of A' of order (n -h)mh^. 12-16] ON DETERMINANTS OF COMPOUND SYSTEMS 69 The conjugate minor in A has for elements those minors of A of order m which are complementary to the elements of M', and hence all those which have Ah as a minor. This is precisely the determinant of Art. 13. Whence the theorem can be stated as follows: If An-h is a minor of A of order n - h, and if we form a determinant iM' with all the minors of order n- m of An-h, and then replace each element by its complement in A, we get a new determinant, whose value is M = Ah(n-h-(1)'-h. A (n-h-1),,-h-i 15. If now we forin all minors of A of order n - m (m> h) such that neither all their rows nor all their columns belong to An-h, which in A therefore overlap A,_h or belong altogether to Ah, these form a determinant N of order n, - (n - h)m-h which is equal to A h (n -h —l) -h (-l1), h- (n —h-1),-h First notice that this is essentially different from the theorem of Art. 11, applied to Ah. There the determinant is formed with all the minors of the same order of A with more elements than Ah, and which do not admit all the rows and columns of Al. Here the determinant is formed with rninors of the same order of A with fewer elements than An-h, and which do not admit all the rows and columns An-h. To prove the theorem it is sufficient to consider in A' the minor 17 complementary to M in A'. For N is exactly formed with regard to An-h as the enunciation prescribes; it has n, - ( - h)m-h rows, and by applying to it the theorem of Art. 9, M -- A (n-l1)m —n,,+ (n-lh) -h N.N = MA (n-1) m-(n-h) m-h or, replacing M by its value, from Art. 14, N =A h(n-h-1)~ -h A (n-i1) (n-h-1)> -h. 16. The theorem of Art. 15 may be stated in a different way, which perhaps brings out more clearly its contrast with the proposition of Art. 11. Suppose as before that Ah is a selected minor of A, of order h. 70 THEORY OF DETERMINANTS [CHAP. VI. Let m < h, and construct all the minors of A which are of order mn and contain at least one row and at least one column not composed of elements of Ah. With these minors as elements we can form a determinant of order nm-hm, the value of which is A (h-l)m-l A (n-l)m-l-(h-D7-l n-h ' This is at once obtained from the result of Art. 15 by substituting n - m for mn and n - h for h, and observing that (h- l)h-m = (h - 1)m-l (n - 1)n-m = (n - )m-i. 17. Netto has given a theorem which may be regarded as an extension of Laplace's formula (iv. 5) for the expansion of a determinant as the sum of products of minors. Let a determinant of order (n +n m) and its reciprocal be represented by (ann) (bim) R- (Ann) (Bnn) (Cmn) m(dm) (mn) (l 'nm and let [ an I be denoted by An. Let k integers mn,, n2,... mk be chosen so that m1 + n2 +... + mk = n. If we choose minors of An of orders ml, m2,... mk formed out of elements in the first mi rows, the next m, rows, etc. and denote them by 8,n, 8m,... 8, Laplace's theorem gives An = + M, m *... ôk. Now m, is obtained from An by the suppression of (n - mi) rows and (n - ni) columns; let Ami be the minor of A obtained by the suppression of the same rows and columns. Then the theorem in question is that -~ Alm A2,... m7 = AADk-1, where Dm = I d,, the complement of A, in A. To prove this, let us, in the first place, apply Laplace's theoremn to expand the reciprocal of An. Thus A n-1' = + ~8 ',...8' 16-18] ON DETERMINANTS OF COMPOUND SYSTEMS 71 where 8'mi is the conjugate of m,, in the reciprocal of A,. But if _n-mi is the complement of 8<, in An Î/ - 3m A mi-l. consequently A.-1 = A (-im) Z + 8n-ln-2... n-mk, whence Z + 3n-m, în-n,2... n-mk = A -n. Next we observe that by Art. 5 AmAn-mi-1l= A' n-m, the complement in R of the conjugate of Am,. Now this is a minor of [Ann and by applying the preceding lemma to \IA instead of to A, we obtain nm n-2... A n-mk Ann Ik-l =-A (n-i) (k-1) D k-i Substituting for 'nmi its value given above, and making use of the relation ' (n m 1) = k (n- 1)-n = (n-1) (k —1)-1, we have finally A- A...k = A Dk-1 as stated. This proof of Netto's theorem is due to Pascal. 18. The theoremn of Art. 13, in its simplest case, may be used to prove an important identity discovered by Kronecker. Suppose we have a determinant of order 2h + 1 (ahh) (bh, h+i) (Ch+i, h) (dh+l, h+) the elements of which are all independent. Let I ah I = A, and let A^ be the determinant of order (h + 1) obtained by bordering A with elements taken from the row and column to which dA, belongs. That is to say, A I = an a12... ah b |~...................... ahi ah2. ah h bhh CÀ CÀ2 z... Ch d,.i 72 THEORY OF DETERMINANTS [CHAP. VI. Then A1 n-dLd A, A11 - d12A,... A,h+A -, *, A,h+lA4 = 0 A +,l~ - dh+4,1 A, A^1 +l2 - dh+i, 2,... Ah+l, l+l - dh+l,h+lA identically. The proof is almost immediate. By Sylvester's theorem | A_16+1, 76+| = DAh; now suppose all the elements dik to become zero, then D vanishes, as we see by expanding according to minors of the last h + 1 rows. But since Ak-didkA is independent of the coefficients dik this expression is in fact the value of Aik when all the coefficients dik vanish. Hence IAk -dA =0, (i, k=, 2,...h+ 1) and this is an identity on account of the independence of the (3h2 + 2h) quantities which it involves. The method of proof here followed is due to Frobenius. 19. There are various interesting theorems about determinants derived from two other independent determinants. One of these, due to Kronecker, is the following. Let A =la,, B =bm 1, and let us form all the rnn22 products aj bhk; then with these products as elements we can construct a determinant C = c ( = mnn) of order mn, in such a way that all tlie elements ajbh, with constant i, h are in the same row, and all those with constant j, k are in the same column. This being so, the value of C is C = AmB. Without loss of generality we may suppose C to have the form ai, (bmm) a12 (bmm)... ain (bmm) a,, (bmm) c22 (bmm)... a, (bmm) a,,i (bmm) am 2 (bm,,)... an (bn,,) 18-20] ON DETERMINANTS OF COMPOUND SYSTEMS 73 If, now, we form minors of order m out of the first mn rows of C, all those which do not vanish will contain B as a factor. Thus, for instance, the minor formed from an (bm) is alrmB; if we replace the first column by the first column of a, (bm,), we get allmn-al2B, and so on. The same argument applies to minors taken from the next m rows, etc. By the generalised form of Laplace's theorem (IV. 8) we may expand C as a sum of products of n-ads of minors of order qn chosen from the first, second,... sets of mn rows. Since each of the minors vanishes or is divisible by B it follows that C is divisible by Bn. In exactly the same way C is divisible by Am, and the theorem now follows by a comparison of dimensions, and the consideration of the case when ai = b= 1 and all the other elements are zero. 20. We shall now suppose that we have two independent determinants, each of order n, A = a,, |, B= bnn 1, and that for each of them we have forrned the systems of;2 (= n 2) elements discussed in Art. 2; the systems for A being denoted by (p,), (q,,) and those for B by (p',), (q'). We can form two new systems each of /2 elements as follows. If i is a fixed integer the quantities p'ik are minors chosen from a particular set of m rows in B. In the determinant A replace, in turn, each selection of rows m at a time by this fixed selection of rows in B. This will give us / determinants, each of order n, which we shall denote by til, ti,... ti. Again let k be a fixed integer, and in B let those rows from which the quantities p'kj are derived be replaced in turn by each combination of m rows of A; the determinants thus obtained will be called uki, uk2,... uk. We have, then, two new systems (tU>), (z6y~). Then by Laplace's theorem A = phiqhl + ph2qh2 + B = p lq'hl+ + ph2q hi +. tik = il qkl + P'i2 qk2 +... * lm = pmi ql + pm2 q'12 +.., 74 THEORY OF DETERMINANTS [CHAP. VI. whence by Art. 3 tilpnl + ti2p21 +... = ilA tilp12 + ti2p22 +... = p iA............................... and therefore tilttki + ti2k2 = A (p' q'i + p'i2q'k +...) = 8ikAB. That is to say, by compounding the ith row of (t,) with the kth row of (u,,), we get AB or 0 according as i is or is not equal to k. 21. Write now T= \t~\, U= \u1ff1; then, by the result of Art. 20, T= p',. i q. j =A(n-M)w. B(-l)n-l U = IP ' ~ q = A (n-l)m-l. B(n-1l)m (Art. 8). Again let Th = I thh I t= tl 2 th. tl, +l... tl g ooo...................... thi th2.. thh th, h+l * * th 0 0...0 1... 0......................... 0 0...0 0... 1 then, forming the product Th U by rows, and applying the results of Art. 20, ThU= (ABB)h Uh+l,h+l, Uh+i, h+2... Uh+l, Uh~,+2, h+l Uh+2, h+2 * Uh+2, Ub, h+l UJ., h+2... 'F;, = (AB)hU,_ where U,,- is the complement of Uh in U. Substituting its value for U, we have U_-/hTh = A\B12, where X1 = (n - l)rn-h, X2 = (n- l) - h. Thus the ratio of complementary minors of T and U is a power of A multiplied by a power of B. CHAPTER VII. ARITHMETICAL PROPERTIES OF DETERMINANTS. ELEMENTARY FACTORS. 1. A DETERMINANT is a rational integral function of its elements, and as such will have certain arithmetical properties dependent upon the field of rationality to which the elements belong. If the elements are independent and arbitrary symbols, the determinant is irreducible, and so are all its minors. This is easily proved by induction; for the expansion A = a|nn I = ai1Al + ai2Ai2 +... + a(inAin shews that if A is reducible one of its factors must be a common divisor of A11, A12,... Ain. But these are determinants of order (n- 1), each with arbitrary elenents, and distinct from each other; so that if the theorem is true for the order (n - 1), it is true for the order n. Since it is obviously true for n= 1, it is true in general. But the case is different when the elements belong to a field of a more special kind. For instance, if the elernents are ordinary integers, the values of A and its minors are also integers, none of which need be prime. So also, if the elements are polynomials in x with integral coefficients, A and its minors are functions of the same type, and the nature and distribution of their irreducible (or prime) factors require special examination. 2. It will be supposed that in the determinant l annl, or A, the elements are all integral and rational in a certain defirilte field; then the same will be true of A and all its minors. It will be further assumed that every integral quantity in the field d can 76 THEORY OF DETERMINANTS [CHAP. VII. be uniquely resolved into a product of prime integral factors. It follows from this that a given set of integral quantities possess a definite highest common factor, which can be found, as in ordinary arithmetic, when the resolution of the given quantities into prime factors has been effected. For the present it will be supposed that the value of A is not zero: it follows that its minors of order ni (<n) cannot all simultaneously vanish. 3. Consider, now, all the minors of A which are of order o. They will have a certain highest common factor; this will be denoted by D,. Altogether, we have n integral quantities D1, D2,... Dn, which we shall call the determinant factors of A. In particular, DI is the highest common factor of the elements of A, while Dn is A itself. The determinant factors are all different from zero. We, shall now prove that D, is divisible by Dol. This follows from the fact that, when any minor of order - is expanded according to elements of one of its rows, each element is multiplied by a minor of order (o - 1), and this is divisible by Do-1. We shall write Da/D_,= -E. With the convention that Do = 1, we thus obtain a new set of integral quantities El 1,... En, which will be called the elementary factors of A. It will be noticed that E, =Dl, and that A=E1E2... En. 4. It is clear that no prime can divide D, which is not also contained in A. Let pla be the highest power of the prime factor p which is contained in D,. Theh since D, is divisible by Dol it follows that in In_-l >. * * 1, so that if we write with the convention that 1, = O, we have the integers el, e2, e3,.... en 2-5] ARITHMETICAL PROPERTIES OF DETERMINANTS 77 none of which is negative, while el + e2 +... + e = +l. In this notation El == Ipe., the product extending to all the different prime factors of A. The values of D,) and E, are not affected by interchange of rows or of columns in A; for this does not alter the values of the minors of order o- (except possibly in sign, which is immaterial) but merely produces a permutation amongst them. 5. If p is any prime factor of A there will be at least one minor of order o which is divisible by pla but by no higher power ofp. Such a minor is said to be regular with respect to p. We shall now prove by induction the following three important propositions:(i) Every minor of order cr which is regular with respect to p has at least one first minor which is also regular. (ii) Every regular minor of order (- - 1) is a first minor of at least one regular minor of order o. (iii) The indices ei associated with the prime factor p satisfy the relation e, >- e,-. First of all we choose a non-vanishing minor of A which is of order p (> 2); let this be M= ah! (h=Ah, h2,... hp; i='i, 2, *.. ip). Let pX be the highest power of p which is contained in all the minors of M which are of order or. Choose a minor of M, of order (p - 2), which is divisible by pXp-2 but by no higher power; and let this minor be called T. Then (vi. 5, 6) we have an identity MT= PS- QR, where P, Q, R, S are minors of M of order (p - 1). Now MT is divisible by pXp+Xp-2 and by no higher power of p, while PS- QR is certainly divisible by p2xp-l and possibly by a higher power; hence Xp +- p-2 Z 2Àp_, 78 THEORY OF DETERMINANTS [CHAP. VII. and therefore Xp - Xpl_ >_ xp_ - Xp-2. Assuming that proposition (iii) has been proved for all values of ar up to (p - 1), it follows that Xp - XP-_ > Xp_ p - Xp-2 >... >. X2 - Xi > X\. Next we take any minor of A of order (p - 1): let this be L = ajk (,j =j, j,... jp-2; k =,,...; k = -). From this we can construct, as in VI. 18, a determinant Lhi by bordering L with elements of M taken from the row and column in which ahi occurs. Then by Kronecker's theorem (l.c.) \ Lahi-Lhi = O (h=hi, h2,... hp; i= il, i2,... ip), identically. Expanding in powers of L, we have LPM=LP-~lM,+LP-2M...+M2M.........(1), where Ma is a homogeneous function in the quantities Lhi, with coefficients which are minors of M of order (p - r). Suppose, now, that pl is the highest power of p contained in L, and pl' the highest power contained in the highest common factor of all the determinants Lhi. Then LP'-M, contains p to at least the power r0, where 7a = (p —)+ Io + Xp,_ (< =l, 2,... p), while LPM contains p exactly to the power o=pl + Xp. Hence +1- T- = (1' - l) - (X- - p-a-i) (I-1)- (XP - Xp-), by what has been proved and assumed with regard to M. It follows from this that '- I x - xP-........................(2), because otherwise we should have p > Tp-l >... > 71 > 70, and then every term on the right of (1) would contain p to a higher power than the term on the left, which is impossible. The relation (2) may also be written X + > p + '........................ (3); 5] ARITHMETICAL PROPERTIES OF DETERMINANTS 79 now if ptn is the highest power of p contained in all those minors of A of which L is a minor, and which are of order p, l' > and consequently Xp +1 > Xpi + m........................(4). We will now suppose that L, M are regular with respect to p. Then l=l p_, Xp = 'p, and the relation (3) becomes Ip + lp-l > Xp-i + I'. Now I > lp, X —l > Ip_, because minors of L, M are also minors of A; so that the three conditions, taken together, lead to l =l, XP-l_ =,p- (p= 1, 2,... p). We have proved, then, that if l = p_ then 1'=Zp; this is equivalent to proposition (ii). It has also been proved that if Xp=lp, then Xp-_=lp-_; this is equivalent to proposition (i). Finally, the relation Xp - Xp-l > Xp-1 - xp-2 previously established for M, becomes Ip- 1p-l > Ip-l- lp-2, or, with a change of notation, ep ~ epi, which is proposition (iii) for o = p. Since each element of A is divisible by pI', every minor of order 2 is divisible by p211, so that which proves (iii) when o-= 2. All that has been assumed for the purpose of the induction is the truth of proposition (iii), as applied to M, for - = 2, 3,... (p - 1); thus the demonstration is complete. From the supplementary relation (4) we infer another proposition, namely:(iv) If Sp and Sp-_ are any two minors of A of orders p and (p - 1) respectively, then SpSæp_ is divisible by the product of the 80 THEORY OF DETERMINANTS [CHAP. VII. greatest common measure of all minors of Sp which are of order (p - 1) by the greatest common measure of all those minors of A of which Sp_ is a first minor. 6. It has been supposed hitherto that the value of A is not zero. We shall now remove this restriction. Unless every element of A is zero there will be at least one complete system of minors of the same order which do not all vanish; while if each minor of order m vanishes, every minor of higher order will also vanish. Thus there will be a definite integer r such that not all the minors of order r vanish, but all the minors of higher order do vanish. This number r is called the rank of the determinant A. If all the elements of A are zeros, it is convenient to say that A is of rank 0. When A is of rank r(< n), the modification which has to be made in the foregoing theory is to put Dr+1 = D+2 =...D = DIn = 0, Er+l = E.+2 =... = En = 0. Every prime factor of Dr will be associated with r finite indices el, e2,... er such that er > er- >.. > el, and if we put el e +... + e. = r,, this is the index of the highest power of the prime contained in Dr. All the propositions of Art. 5 remain true, and the proof of them is still valid; in the enunciations, however, it must be understood that p or o, as the case may be, does not exceed r. 7. The term elementary factor (or elementary divisor) has not always been used by different writers in the same sense. Sometimes it means one of the quantities E, sometimes one of the powers pea. Frobenius refers to E, as the uth elementary factor of A, while by an elementary factor he means one of the powers pe. It may be convenient occasionally to refer to pe, as an elementary power-factor, or simply a power-factor of A: but as a rule we shall follow Frobenius's terminology. Of course the 5-8] ARITHMETICAL PROPERTIES OF DETERMINANTS 81 system of the quantities E, determines the quantities pea and conversely. 8. Suppose that A = l ann and B= b nn are any two determinants of the same order, with elements all belonging to the same field. Let their product be expressed, in one of the usual ways, as a determinant C= j cn, of the sane order n. We shall now prove that the oth elementary factor of C is divisible by the oth elementary factor of each of the components A, B. To fix the ideas, assume that the product AB has been formed by compounding rows of A with columns of B. Let Bp, C, be two corresponding determinant factors of B, C and let /,, y, be the exponents of the highest powers of the prime p contained in them. Since every minor of C can be expressed as a linear function of minors of B it is clear that 7y, >, / and consequently Cp is divisible by B,. What has to be proved is that 7P - p-i - > p- p-l Take L a regular minor of B of order (p - 1), and M a regular minor of C of order p. Let us write L = bjk (j =jl, j, * jp-i; l=k1, k2,... kp-), M= I Chi (h =h, h,... h p; i= il, i2,... ip). We proceed to apply to these minors exactly the same argument as that applied in Art. 5 to the determinants there denoted by L, M. Instead of the determinant there denoted by A we shall now use a determinant of order (2p - 1) which may be briefly written in the form A (L) (P) (Q (M where (P) is a matrix taken from B, with suffixes (j, i,)) (ji, i2),... (jil ip) ( jp-i1) i( jp- 2')) *.** (jp-.. 9p), and (Q) is a matrix taken from C, with suffixes (h k, (h k),... (h1, kp-) (hp, ki), (hp, k2),... (h., kP-I). S. D. 6 82 THEORY OF DETERMINANTS [CHAP. VII. Instead of the determinant Lhi of Art. 5 we obtain one which consists of the new L bordered in such a way that the last row is Chk, k h,c2 Ch, k, kp_l Ch, i while the last column, read from the top, is bj b,i,... b,. If, now, pl' is the highest power of p contained in all these last determinants, the formula (3) of Art. 5, applied to the present case, gives yp + lep-l > yp-l + l. But each of the determinants corresponding to Lhi can be expressed as a linear homogeneous function of minors of B of the order p; hence 1' > /3 and consequently p + p-l 3> p-l + P/3 or 7Pp-1 p-i p- lp-P. This proves that every elementary factor of C is divisible by the corresponding elementary factor of B. By similar reasoning it may be shewn that each factor of A is contained in the corresponding factor of C. The theorem may be obviously extended to the case when C is compounded of three or more determinants of the same order and belonging to the same field. 9. A very important special case is when the value of one of the determinants A, B is unity. Suppose that A = 1; then the elementary factors of C are identical with those of B. To see this, let E, and E,' be any two corresponding elementary factors of B, C respectively. Then, by the theorem of Art. 5, E,' is divisible by E,. But since (A) is a unit matrix, we can find another matrix (A), with integral elements, such that (A) (A) =[1], and hence (A) (C) = (A) (A)(B) = (B), identically. Therefore E, is divisible by E,', and finally E.J = E,. Two determinants of which one can be identically transformed into the other by compounding it with a unit determinant are 8-10] ARITHMETICAL PROPERTIES OF DETERMINANTS 83 often said to be equivalent. Thus equivalent determinants have the same elementary factors, and therefore also the same determinant factors. 10. It will now be supposed that the elements of the determinant A are ordinary whole numbers. We shall prove that it is possible to find two unitary matrices (P), (Q), such that (P) (A) (Q)=(E)=[lE], where (E) or [En] denotes a matrix of which the principal diagonal consists of the elements E1, E2,... E, (the elementary divisors of A) while all its other elements are zeros. Consider the effect of transforming any determinant by one of the four following elementary operations:(1) Interchanging two rows; (2) Interchanging two columns; (3) Adding to any row k times another row, where k is any integer, positive or negative; (4) Adding to any column k times another column. Any one of these transformations leaves unaltered the values of the elementary factors E,, because the complete system of minors of any order for the new determinant coincides, except as to arrangement, with the same system for the original determinant. Moreover, by the rule for the composition of matrices (v. 2) it is easily proved that if D' is the determinant derived from D by either of the operations (1), (3) there is an identity (') = (P) (D) with IP = + 1 or - 1. Similarly, if D" is derived from D by one of the operations (2), (4), there is an identity (D") = (D) (Q) with I Q=+1 or -1. In the given determinant i a,, I, unless every element is zero, there will be at least one element of which the numerical value is equalled or surpassed by every other element which does not vanish. By the application of one or both of the operations (1), (2) this element can be brought to the head of the principal diagonal. 6-2 84 THEORY OF DETERMINANTS [CHAP. VII. By adding multiples of the first row to the other rows, we can reduce every element in the first column, except the one at the top, to a value which is either zero, or numerically less than the value of the element at the top. We now interchange rows until the first column is headed by an element, not zero, of the smallest numerical value; and then proceed, as before, to reduce the other elements of the first column. This process must eventually come to an end; so that, after a finite number of transformations, we get a first column in which every element is zero except the one at the top. We now operate upon the transformed determinant so as to reduce the values of the elements in the first row by means of the elementary operations (2) and (4). Ultimately we get a first row of the form a,, 0,... 0 and a is certainly not greater, numerically, than the number previously in the same place. If the elements now below a in the first column are not all zero, we reduce them, as before, by the operations (1) and (3), until we have a column with a' at the top, and all the other elements zero. The elements a, a', etc., which we thus get at the head of the leading diagonals, diminish continually in numerical value, so that a stage must conie when no further reduction is possible. When this is so, every element in the first row is zero except the first and at the same time every element in the first column is zero except the first. Thus A has been transformed into A', where A'= 77,, O,... O O, b22, b23,... b2, O, b32, b33,... b n.......o............ O, bm,n bn3... bnn and r is different from zero. Suppose, now, that there is an element bik which is not a multiple of: then we add the (k + 1)th row of A' to the first row, 10] ARITHMETICAL PROPERTIES OF DETERMINANTS 85 and carry out the process of reduction, as before, until we get another determinant like A': this will have, instead of 7, an element of less numerical value. Here again the process must come to a stop after a finité number of operations: so that we ultimately get a determinant A' where every element bik is a multiple of V. If we now put?î = el, bik = eCik, then A'= el, 0,,..., e122, e1c22, e,.. elC2n 0 1c32, e c,, e,... ec. 3........................... O; e, cn2, eCn3,,,. e Cnn and if we put -Cik (z g =, 3,...n), the determinant C can be treated in exactly the sane way, so as to transform it into C'= e, O, 0,... O 0, e2d33, e2d34,... e2d3n 0, e2d43, e2d44,... e2d4, 0, e24d3, e2d,4,... e2dnn and there will be a corresponding change of A' into el, 0, 0,... O 0, ele2, O,... O 0, 0, c,33,... n '......................... 0, 0 n, n..3 * nn where each element aik is a multiple of ele2. By continuing the argument we arrive at the final result that (A) can be transformed into the equivalent diagonal system [E1, E2,... En] where E, = ee2... e,, and el, e2,... e. are integers. It has been assumed that some one element of A is not zero 86 THEORY OF DETERMINANTS [CHAP. VII. this being so, e, is certainly not zero. But the determinant above denoted by C may have every element zero: in this case we put e2 = e. =... = en = 0. In like manner, after getting finite integers el, e2,... er the determinant which at that stage corresponds to C may have all its elements zero: we then put er+l- = e+2-... = en = 0. 11. The reduction of (A) to the normal form (E) having been effected by the four elementary operations, we have au identity (E) = (Ph) (Ph-,)... (P,) (A) (Q) (Q2)... (k), and since the composition of matrices is associative this may be put into the form (E)= (P) (A) (Q), where (P), (Q) are unit matrices. Hence the cth elementary divisor of (A) coincides with the o-th elementary divisor of (E). But it is easy to see that the latter is E". For, in the first place, the non-vanishing minors of (E) of order ao are simply the products of El, E2, etc. taken o- at a time; and of these the greatest common measure is D, = EE2... E,. Hence the oth elementary divisor is D,/D_-, that is, E,. It follows that two systems (A), (B), of the type here considered, which have the same set of elementary factors must be equivalent. For if (E) is the diagonal system of elementary divisors there will be four unit matrices (P), (Q), (R), (S) such that (E) = (P) (A) (Q)= (R) (B) (S), and hence (B) = (R)-1 (P) (A) (Q) (S)-', where (R)-~ is the unit matrix such that (R)-' (R) = [1], and (S)-l is similarly defined. 12. The argument and results of Arts. 10, 11 apply, mutatis mutandis, to the case when the elements of the determinants 10-13] ARITHMETICAL PROPERTIES OF DETERMINANTS 87 considered are not integers, but rational integral functions of one or more variables. In fact the process of reduction to the normal form (E) can always be carried out when the greatest common measure of any two of the elements can be found by a process of chain-division analogous to that which is used for two ordinary integers. 13. The elementary divisors of a determinant may also be regarded as being associated with its matrix; and the theory may be extended to rectangular matrices in general. All that is necessary is to enlarge the matrices by adding rows or columns of zero elements until they become square. CHAPTER VIII. DETERMINANTS OF SPECIAL FORMS. 1. WHEN a square array is written down, it is natural to inquire what simplifications arise in the determinant of the array when special relations are supposed to exist between the elements. And looking at the figure the relations which naturally suggest themselves are those which depend on the geometrical form which the array assumes. Hence we have various forms of determinants obtained by supposing relationships, of equality or otherwise, to exist between elements situated symmetrically in the figure; this shews how the notation employed has influenced the development of the theory. The most important of these special forms are symmetrical and skew symmetrical determinants. Here the special form of geometrical symmetry considered is with regard to the leading diagonal. Eleinents which are situated in regard to the diagonal in the position of a point and its image with respect to a mirror coinciding with the diagonal, have been called conjugate: two such elements are denoted by aik and aki. 2. If aik = aki, the determinant is called symmetrical. The square of any determinant may be expressed as a symmetrical determinant of the same order. For | a^ 12 = Cik where ik = ai, akl + a2 ak2 +. * = Cki It follows from this that every even power of a determinant is a symmetrical determinant. 1-3] DETERMINANTS OF SPECIAL FORMS 89 3. We may also suppose the determinant to be symmetrical with respect to the centre of the s e formed by the elements of the determinant. Two cases arise, according as the determinant is of even or of odd order. First, if the order of the determinant is 2r, we may write it in the form: D== ai, b1, c1... mi, ni, vi, 1i... 7, /i, ai a2, b2, C2.. m2, n 2, 2... 72, 2S, a2................................................ ar, br, Cr... mr, br), Vr,,r *.. 7r, ATr, ar ar, f3r, 7r -** ) Vr, nr, m,... Cr, br, ar a2, 32 72, Y2.../, r2, n2, qM2.. c2, b2, a2 aL, 3i,.1..., 1, i, i, m... c, b1, a1 In this determinant add the last column to the first, the last but one to the second, the (r+ 1)st to the rth, then it becomes D = 4 a + a, bi + 31... n2 + vi,, Z L,... /3, al a2 + a2, b2 +/2. *2 + +2, V2, IL2... 1 *, 2a..o..............*.**.*.*.....****.**..*******.... a. +arr, br +/... n + Vr, r, /Ir...el., ar ar + ar br +/... nr+ Vr, n, mr... br, ar................................................... a2 + 24, b2 +/2... n 2 + v2, n2, m2... b, a2 al + ac, b1 +31... n + v1, ni, mi... b, al Now subtract the first row from the last, the second from the last but one, the rth from the (r + l)st, then D= al+a1, b+fi+...n- v,,, /... i, a1 a2+a2, b2+2...2+v2, v2, /2..*., a2 ar + ar, br+fr... *nr + Vr, r,, Ur... r, ar 0 O, 0..- y, mr '- -hr... br-/3r, ar-Otr 0, 0... 0, 2-v, m 2-2... b2-/2, a2-a2 0 O.. O, n, - v, m-0 -..., -,..., ai-a1 90 THEORY OF DETERMINANTS [CHAP. VIII. Hence (iv. 7), D= a~l+al... fl +t r - rr... ar - (ar.D -- a........... ar + ar... nr + vr ni - vl... a, - ca, But if the order of the determinant is 2r + 1, it may be written in the form D= a,, b1... n, ui,.... l, a a2, b2... n2, 2...12, a2.........*.....******************* a. b., ur,... r, ( cir V1, V2... Vr, p, Vr... V2, V ri,.r... v r), Ur, nr... br, ar aI, 31... v1, u1, n,... b1, ai By proceeding exactly as in the former case, we can shew that D = ai + a... nl + vl, Ui nr- r... ar-ar ar + ar... nr + l'r, Ur ni - V... aL -ai 2v1,... 2v,., p So that when a determinant is symmetrical with respect to the centre of the square formed by its elements, it reduces to the product of two other determinants. 4. If in a determinant the conjugate elements are equal in magnitude but opposite in sign, i.e. if aik = - aki, the determinant is called a skew determinant. If, moreover, ai = 0, the determinant is called a skew s metrical determinant. 5. It will be useful to notice the connexion between two minors of these systems, such that the rows and columns suppressed to obtain the one minor correspond to the columns and rows suppressed to obtain the other. Two such minors may be denoted by P = af, apg..., Q= afp, afq... aqf, aqg... a9p, agq...,,.oo........ o o............. 3-8] DETERMINANTS OF SPECIAL FORMS 91 6. If the determinant is symmetrical, i.e. if aik = aki, clearly P = Q. A special case of this is, that in a symmetrical determinant Aik = Aki, for Aik is got by suppressing the ith row and kth column, while Aki is got by suppressing the kth row and ith column, thus these determinants are of the same nature as P and Q, and are therefore equal. Thus the determinant of the reciprocal system is also symmetrical. If A is the determinant of the system, dA Aik + k daki daik - k daik = 2Aik. But d Aii daii In a symmetrical determinant Aii and the like are still symmetrical determinants. 7. If in Art. 5 aik = -ak, we see that P= apf, apg... = -afp, -agp... (-1) Q, aqf aq... -afq, - agq... m being the order of the minors. Thus if m is even P=Q, but if m is odd P =- Q. 8. The calculation of skew determinants reduces to that of skew symmetrical determinants, which we shall therefore now consider. A skew symmetrical determinant of odd order vanishes, for if we multiply each row by - 1, since aik = - aki, this changes the rows into columns, which does not alter the value of the determinant. Hence, if n be its order, A = (- n A; and therefore A = 0 if n is odd. 92 THEORY OF DETERMIINANTS [CHAP. VIII. The minor Aik differs from Aki by the sign of every element; hence A ik - ( 1)n- Aki Thus Aki= Aik if n is odd, but = - Aik if n is even. Thus the reciprocal system is skew if n is even, but symmetrical if n is odd. Ai is a skew symmetrical determinant of order n - 1, and hence vanishes if n is even. We have dA A A daki dj- Aik + -Ak daik daik = Aikc - Aki = 2Aik if n is even = O if n is odd. 9. A skew symmetrical determinant of even order is a complete square. For if A = t aik is the determinant, A,, vanishes because it is a skew symmetrical determinant of odd order. Hence (vI. 6), if aik is the complement of aik in A11, I i ik -, or aikk=ik2, U ki, s kk since aik = ak (Art. 8). Now by (iv. 24) if we expand according to products of elements in the first row and first column, since A1 = 0 A = - Tai ak aik, where i, k take the values 2, 3... n; or1,)- A-= ai alk C Voii akk = {Za V/a-12. ThIus A is the square of a linear function of the elements of a row. N'ow aii is a determinant of order n - 2, which is even if n is even. Thus a skew symmetrical determinant of order n will 8-11] DETERMINANTS OF SPECIAL FORMS 93 be the square of a rational function of its elements if one of order n - 2 is so. But when n = 2, 0, a12 = a122. a21, 0 Thus skew symmetrical determinants of orders 4, 6... 2r are squares of rational functions of their elements. 10. Since if n = 2 the square root contains one term, when n =4 the square root will contain 3, when n= 6 it will contain 5.3 terms, and so on. Hence a skew symmetrical determinant of even order n is the square of an aggregate of 13..5... (n-l) terms, each consisting of the product of 2n elements of A. In particular al2a34... an- is a term of </A, for n (al2a34... an-)1)2 = (- 1)2 a12. 34 a n-, a2 a43... ann-i 11. This function /A is of importance in analysis, and has been called a Pfaffian by Prof. Cayley on account of the use made of it by Jacobi in his discussion of Pfaff's problem. That value of V/A which contains al.2a4... an-_ as first term with positive sign will be denoted by P=[1, 2...n]. The remaining terms of P are got from the first term, a12 a34... an-ln by interchanging all the suffixes 2, 3... n in all possible ways, and giving a sign corresponding to the number of inversions. Since aik = - aki it is possible to effect the interchange in such a way that ail the terms are positive. The Pfaffian changes sign on interchanging only two suffixes i and k. For if we interchange i and k in the determinant, this interchanges the ith and kth rows as well as the ith and kth columns, thus the value of the determinant remains unchanged. If P1 is the new value of P, P2 = P2 Hence P1= + P. 94 THEORY OF DETERMINANTS [CHAP. VIII. To determine which sign we are to take, let us consider the aggregate of terms aikpik which contain aik. Then pik only contains terms whose suffixes are independent of i and k. The corresponding aggregate for Pl is akipik, which, in consequence of the relation aki= - aik, proves that P, =-P. 12. The minor aii is also a skew symmetrical determinant. We shall shew that ^/aii( —1)i[,... i-1, l-l,... n], or with i- 2 cyclical interchanges ii = [i + l,..., 2... i - 1]. Since ik = Oii akk, it follows that the terms of the product V/ca /akk are either equal to those of cik, or equal with opposite signs. Now the product (- I)i+k [2... i-l, i+l... n] [2... k-, k+l... n] and the determinant ck = a2,......a2, k-, a2, k+.... (- )i+k,.................................... ai-l,2. * ai-l, k-l, ai-,, k+ *. ai+,, 2 * ** ai+l, k-1, ai+1, k+l *.* o.................................... by the same number of interchanges of two suffixes, become respectively [k, p.q, r, s.. u, v] [p, q,, s... v, i] and akp, akq, akr *~. aki app, apq, apr... i.......................... avp, avq) avr... avi And the term akp aqr... auv ~ apq ars * * avi of the product agrees in sign with the first term of the determinant akpapqaqr... avi, whence the theorem follows. This proposition serves to determine /au, V/a22 as functions free from ambiguity of sign. 11-14] DETERMINANTS OF SPECIAL FORMS 95 13. Since we have shewn in Art. 9 that JA = ai.22+ a13 + 3 3... + ain Vnn it follows that [1, 2... n] = a2, [3... n] + ai3 [4... n, 2] +... + aln[2... n- ]; a relation which enables us to determine Pfaffians of order n from those of order n - 2. Observe that after we have selected the suffix 1, the others are written cyclically. Hence [1, 2] =a12 [1, 2, 3, 4] = ai2a34 + a13a42 + a4a,23 [1, 2, 3, 4, 5, 6] = ai, [3, 4, 5, 6] + a,3 [4, 5, 6, 2] + a14 [5, 6, 2, 3] + a[6, 2, 3, 4] + a6 [2, 3, 4, 5] = ai2a34a56 -+ a12a35 a64 + a,2a36 a45 + a3 a4 a62 + a13 a46 a25 + a3 a42 a56 + a14 a6 a 3 + a4a52 a36 + a14 a3 a62 + ai56 a6 a34 ai, a63 a42 + a5 a64a23 + al6 a23 a45 + al6 a24a 53 a6a, a2534. In particular O, a, -b, c =(ad+be+cf)2. -a, O, f, e b, -f, O, d -c, -e, -d, O 14. In a skew symmetrical determinant of even order, Aii vanishes, being a skew symmetrical determinant of odd order. But (Art. 8), dA Aik = 2 dai A daik -2 d~ik d:[11, ]...]a n]. Now P=[l1, 2... n] = (- l)i-[, 1...i-, +... n] =(-)i-l {ai [2...i-, i+1...n] +... +ai(-I)k-1 [1, 2... -1, i+1... k-1, k+1... n]+..}; 96 THEORY OF DETERMINANTS [CHAP. VIII. hence A = (- l)ik [1, 2... n] {ik}, where {ik} is the Pfaffian got by omitting i and k in [1, 2... n]. 15. In a skew symmetrical deterininant of odd order Ai is a skew symmetrical determinant of even order, and is hence the square of a Pfaffian; viz. Aii = [1... i - 1, i 1... n]2, /Ai = (- 1)i-l 1... i-), i + 1... n] =[+l... n, 1...i-l]. Also, since A = O, Aik2 = AiiA k. IHence Aik= [i+ 1... n, 1... i - 1] [ + 1... n, 1... k-]. 16. The result of bordering a skew symmetrical determinant is also of interest. The result assumes different forms according as the determinant which we border is of odd or even order. Let the original'skew symmetrical determinant be A = | ik, and let the bordered determinant be D -aa, a3, aal, aaaa3., a,,O a, a.., a,a... at2, a21, a22, 23 a3, a31, a32, a33 By Cauchy's theorem (III. 24) D = apA A - ai ak Aik. Now, if A is of odd order it vanishes, and Ak =[i+1... 1...i-, 1... -1]; hence, if we suppose that aPk = - ak, D = Ea,,k[i +1...n, 1... i-][k+1... n, 1... k-] = (a.L [2, 3...n] +...)(a [2, 3... ) + - [a, 1, 2... n] [, 1, 2... n], 14-18] DETERMINANTS OF SPECIAL FORMS 97 where in the Pfaffians such expressions as ai,, ak which do not occur in the determinant are supposed to mean - ai, - ak. But if A is of even order, D = a [1, 2... n]2 + aai Ck(- 1)i+k (ik} [1, 2... n] (Art. 14) =[1, 2... n][ a,, 1, 2... n]. 17. We have hitherto treated of skew symmetrical determinants: it is easy to reduce to these the calculation of skew determinants. Namely, by IV. 23 D' = D + aiDDi + a kkD + a k * + ai, a..... ann, where D is what D' becomes when all the diagonal elements vanish. Di is what the coefficient of aii in D' becomes when the diagonal elements vanish; Dik the coefficient of aiiakk in D' with the elements in the leading diagonal zeros, and so on. If all the elements in the leading diagonal are equal to x we can write this D' = n + xn —zD2 + xn-4D4 +... + xn-mDm +.. where Dm is a minor of order m got by suppressing n - m rows and columns which meet in a diagonal element, the other diagonal elements being put zero, and the summation extends to all m-ads in n. If ni is odd, D,, vanishes, and if m is even it is a complete square. Thus, the elements being skew, x, ail ai3 = x3 + x (a22 + a231 + a223) a21, x, a23 a31, a32, x X, a2, ai3, al4 = 4 x+ Wx2 (a12 + a213 + a1 + a223 + a2 +- a34) a2, a, a2, 24 + (a1a34 + al3a42 + al4a,3)2. a31, a32,, a34 j a41, a42, (143, 18. We can apply this last theorem to prove Euler's theorem concerning the product of two numbers, each of which is the sum of four squares. Namely, we have S. D. 7 98 THEORY OF DETERMINANTS [CHAP. VIII. a, b, c, d = (a2 + b + c' +d2)2 -b, a, -d, c -c d, a, -b -d, -c, b, a p, q, r, s = (p2+q2 +- + 2)2. - q, p, -s, r -r, s, p, -q -s,-r, pq, p Now multiply these two determinants by rows, then if we write A = ap +bq +cr +ds, B=- aq + bp - cs + d~r, C=-ar+bs+cp-dq, D=-as-br+cq+dp, we get a skew determinant of the same formn as the other two, whose value is (A2 + B2 + 02 + 2)2, whence (a2 + b2 + c2 + d2) (p2 + q2 + r2 + s2) = A + B2 + C' + D2. If we were to effect the multiplication by rows and columns we should get another form of the same theorem; by suitably permuting the rows and columns we get still further representations of the way in which the product of two numbers, each of which is the sum of four squares, can be represented as the sum of four squares. 19. We have seen that the square of any determinant is a symmetrical determinant (Art. 2). Cayley ald Brioschi have shewn independently that the square of a determinant of even order can be represented by a skew symmetrical determinant of even order. Brioschi's method is as follows: We have A = -a1, a12... a12_-, ain =- ac2, - a11... Clt,, - i-la21, a22 a... 2-1, a t2. n a22, - t21 *.. a11, - t.2ln-1 anl an2 * a~,nn-i, ann' a nn, - n ail1 —,?'nn — 18-20] DETERMINANTS OF SPECIAL FORMS 99 Multiply these two equal determinants together by rows, and we obtain: A2= Oj /, 1, 1... 3l4 0l O is 121 I, 0, 23* *.........oooo........... l i 2, 1112, 113... o where = a.,1a2 - a2 (tl + a,3l a34 - a,'4a3 +... + a,., a,,sn - a,,,na,,-,, so that Iss = O, ls + sr = O. Thus A2 is represented as a skew symmetrical determinant. It follows that A can be represented as a Pfaffian of the functions 1. If n = 4, for example, ai,... a14 1= 12134 + 113 142 + 114123 a41... a The sign is determined by making the sign of a single tern in the determinant and Pfaffian agree. If, instead of interchanging colunins, we interchanged rows, we should get another independent representation of the determinant as a Pfaffian. 20. A third class of determinants comprisés those of the form D=- a, a2, as... an a.2 a3, a4... a,+l a, a, a, a a3, a4, aO+... an+2 a ' a n ) ~Ctn+ ~ n+2 *a * a.n-1 where all the elements in a line at right angles to the leading diagonal are the same. If the elelnents had been written with double suffixes we should have had the relation apq - ars whenever p + q = r + s. Such determinants have been called orthosymnmetrical. Their most important property is that we can replace the elements by differences of a1. 7-2 100 THEORY OF DETERMINANTS [CHAP. VIII. For if we operate on the rows as we did in Chap. III. 5 (iv), and put A(x = a+i- a., &C. 1 - =-. a1, a.2,... aCn 1 Ai q a, Aa,... Aa A2C1, l2a2a,... ~2an............................... A\i,-l At-la2,... A1n-lan Now repeat the same series of operations on the columns, beginning at the last, then D=I ai, Aal,... An-la al, A2al,... Ana^ i A2a1, A3al,........................ \?n-lal, nal... AS2f-2 a An important example of this class of determinants is that where ak is a function of k of the rnth degree in k, whose highest term has coefficient unity, so that the quantities a1, a2... form an arithmetic series of the rnth order. If m = n - 1 all the eleinents below the second diagonal vanish, while all those in it are equal to (n - 1)!, whence the value of the determinant is 7(4 - 1) (- 1) {(n - )!!}. If mn is less than n- 1 the determinant vanishes. 21. The determinant of order r + 1, Mp wp+l, mrp+2... m?~p+r (m + 1), (m + l)p+, ( + )p+2... (n +1)p+. (m + 2)P, (m + 2)P+l, (m + 2)+2... (1n + 2)+. (m + r)p, (rnm r)p+l, (m + r)f+2... (m + r)+,. m (m - )... (m -p + l) where = - wP= 1. 2.. p ' though not orthosymmetrical, is of a similar nature; let us call it Vrnm,p 20-21] DETERMINANTS OF SPECIAL FORMS 101 Divide its first row by m, the second by m + 1,... its (r + l)th by m + r. Then multiply the first column by p, the second by p + 1,... the last by p + r. Then V in (m + 1)... (m + r), p - (p +l). (p +r) I (m~- l)-, (m - )... ()p - (m-l)p+_i mp_-l,?71p...??1p+l-._ (m + r - 1),-,, (m + r -1)p... (m + r - )p+-lor, if we multiply nurnerator and denominator of the fraction by (r + 1)!, Vm, ( + vr)r+ ''V_,i (,p + r)r+l Thus by giving to m and p different values we obtain the series of equations - (m + r-l1)r+i n-ilp-1 (p + r - ) -,p-2.....................,................... (m + r - p + 1)r+i V V fl-p~1 +(r + 1).+l m-p o Now Vp,o is the value of the last determinant in IIi. 5, when we write m- p for m and 1 for d. Hence its value is unity, which gives, when we multiply the above equations together and cancel like factors, YVm - (m + r),.+l (n + r - 1),+i... (L + r - p + l),+ mp (p + r),.+, (p + r - 1)r+i... (r + 1),+, Another expression can be obtained for the determinant by dividing the first row by m., the second by (rw + 1)p,... the last by (ni + r)p. Then multiply the first column by p0, the second by (p + 1 ), the last by (p + r),.; the transformation gives Vm ( _. n +p (m 1 + l)p (2 +) 2)p... (m + q-), p (p + 1)p (p +2),... (p+r)p A remarkable special case of the first form is when p = 1, the value of the determinant being (rm + r),.+1, i.e. the last element in its leading diagonal. 102 THEORY OF DETERMINANTS [CHAP. VIII. 22. If in the determinant of Art. 20,ak+ = (c + k +)n) (c+ (C + m- )... (c + c + 1) ak+î = (c + k +,n == 1.2. 1.2... mi then if m = n - 1, A'-lc1 = 1, and we have (c + n - 1)_, (c + n)1_... (c + 2n - 2),-1 n (-1) (C +?Z)}1- (C ++)n-i ~. (c+il + 2n- -... 'oo............................o........................... (c + 2n2)n_, (c + n - 1),,_-... (c + 3n-:3)- 23. Another class of determinants consists of those of the form D= a1, a1... an a a1-i C,n...,1-2.1, a.................. ao, a.3... al where the element in the leading diagonal is always a1, and the rest of the row is filled up with a,... a,, in cyclical order. The peculiar property of this determinant is that it divides by Ct1 + C-2) + aC O2 +... + acon —1, where co is a root of the equation x12 = 1. For if A1, A... A,, are the complements of the elements of the first row of this determinant we have (iv. 11) aCA1+ + a.A+. +... + Ca, An, = D alA A+aA1a+... + a1A1i =0 aA, A+ a2A 1 +... + a,, A_- = 0 Now consider the product (ai + ao + aoe ' + a,,o-) (A + A0 + Aa-2 +... + A,-n-1 +). The coefficient of o"k- is Alak + A2ak+l +... + Aak_-. If k is equal to unity this is equal to D, by the first of equations (1), but if k is not unity it vanishes by one of the other equations. Thus D divides by al + a.0 +... + a,, w-1. 22-25] DETERMINANTS OF SPECIAL FORMS 103 Hence D = (ai - a, +. + a.) H (ai + a,2 + ao~ +... + ano-1), where w is one of the roots of the equation n - 1 = 0, unity excepted. 24. Another elegant demonstration of the theorem of the preceding article is the following. If Oli, o2... ~on are the n roots of unity, let P= — 1, o0i, )il2... oln-1 5 1O, (~)2 Û)22 **. -(1)2n1. 1, 02, 0)n2... W^n -1 Then if we write a, + a2i, + a3o)2 +... + Cta. )-l = < (o), and remember that (o = 1, DP = <()l),. (fp)2) (012 (oe01), 022 P (o2) ~-^((o),;-~(û,)................................ = P? (ci) <p (02)... (o),), whence D = + (OI) < (02)... (o). 25. Mr Glaisher has shewn that a determinant, such as that in Art. 23, of order 2n, can be expressed as a similar determinant of order n. Namely a1, a>... a2i A2, A2... An a>2n, Ct-... a A... AC 2n-2 a2n-1, at2fl.. a.2 A;2 -........... ~ ~~ j 2........... a2, a3... a where A = lala - act, a + an-2l..-a. - an A2 = a al1 - a2 a2 + aia... - a4cwrn............................. -....... - 1................................................ 104 THEORY OF DETERMINANTS [CHAP. VIII. For the first determinant = II (a, + aoû + a co2 +... + a2nco2-l), o being a 2nth root of unity; and since for every root o there is a root - o, this = n1 (Al + A2o2 + A3<o4 +... + A,6oe-2), which product is equal to the second determinant. For the 2nth roots of unity being denoted by + 1, ~+ w, ~+ o... **,_, the nth roots of unity are 1, o12,,22... W,_,. For example, if 'n = 2, a6,b c, d = A, B d, a, b, c B, A c, d, a, b b, c, d, a where A = a2 + c2 - 2bd, B = - b2 - d2 + 2ac, and the value of the determinant is a4 - b4 + c4 - d 2ac2 + 2b2d2 - 4a2bd + 4b2ac - 4c2bd + 4d2ac. 26. If in the determinant of Art. 23 we suppose a, ( r-l) +( +r- )+(2n+r- +" =(r-1)!! (2 + - (nr D = exII (a1 + 2ao + a3co2 +... + a,,o'î-l) = ExIl eoX Ex (lt+,l+ot2+..+toî-l) =1. 27. Determi whose eler e imial have been discussed with great minuteness by.eipel, who has given an immense nutnber of theorems relating to this class of determinants. One or two of these we shall now consider. 25-27] DETERMINANTS OF SPECIAL FORMS 105 The value of the determinant nik, t) p n, q q,2... tn k-1 (m +l)k, n + l, (p+l)(n + 1l), (q +l)( + l)2... (t +l)( + )k-i (m + 2)k,n + 2, (p + 2)(m + 2)1, (q + 2) ( + 2)2... (t + 2)(m + 2)k-............................................................... o........ (m +c)k, n + k, (p +-k)(M + k)1, (q + )(z +k)2... (t +k)(m+ +/ )kis (n - n) (mn - p ) ( - 1) q - -2)... (2 - t- k+ 1). We must first shew that the determinant vanishes when mz is equal to any one of the quantities n, p+l, q+2...t+k-1. First let m = n, then the determinant is mk,,, pmn1, q m2.. (n + I)k, (M+, (p + ) (n, + 1), (q + ) (M + 1)... (m~ + k)k, n k + k, (p + k) (m + k)1, (q + k) (ni + k)2... If we subtract the second column, multiplied by p, from the third we see that the determinant is independent of p. Do this, and divide the first row by m, the second by + 1, the third by m +2..., then multiply the first column by k, the fourth by 2, the fifth by 3..., then the determinant reduces to the product of vm (m + 1) (m + 2)... (n + k) 1.2...k and the determinant (m - l)k_, 1, 0, q (n - 1), r (n - l),_k-1, 1, 1, ((q + l),, (r+ ),......................,,.............o......................................... (+ k - )k-i, 1, k, (q + k) (,+ + k - 1), (r + i) (( + - 1)2... Multiply the second column by q (n - 1), the third by q (m- 1)o + 1. m, and subtract their sum from the fourth column, and we get the new determinant 106 THEORY OF DETERMINANTS [CHAP. VIII. 2 (m - 1)k-,, 0, 0, r (m,-l1)... 1 '~rk-_11, 1, 0, (r + 1) q * (n + l)k- 1,,, (r + 2)((m + 1).,..,.........................................., (m + - l)k-1, 1, k1, k2, (r + k) ( + k- 1)... In this determinant multiply the second column by r (m - 1)2, the third by r (rm- 1) + 1..m, the fourth by r (m- 1)0 + 2. mn, and subtract the sum of their elements so multiplied from the elements of the fifth column, and proceed in a similar way with the altered determinant. Finally we reduce the determinant to the product of a finite number of factors and (1 - I)k_-, 1,,... 0, 0 m_-1, 1, 1, 1, 0... 0, (m+ l)k-,, 2, 1... 0,.............................................. (mk + k - l)k-i, 1, k1, k2... Ck-2, /Ik-1 In this determinant multiply the second column by (m - l)k_-, the third by (r - l)k-2, the fourth by (m - 1)k-_, &c., and subtract their sum from the elements of the first column; then each element of the first column, and consequently the determinant vanishes. Hence our determinant divides by m- n. Similarly we can shew that it divides by each of the other factors, hence it is equal to C(,, - ) (n - p - 1) ( - q2)... (2 - t - k + 1), where C is independent of n, p, q... t, because the determinant is linear in each of these quantities. To find the value of C put z= = q...=.t=0; then we get ',k 1, (,m + 1)i, (n + 1)2... 2, 2 (m + 2), 2 (n +2)... 3, 3 (m+ 3)1, 3 (m~2 + 3)2.................................... k, k (m + -)1, (m + k)2... =m (nm - 1)... (m -- k + 1). 27-28] DETERMINANTS OF SPECIAL FORMS 107 But the determinant last written is equal to k!, as we see by putting d = 1 in the last determinant of ni. 5. Hence C= 1 thus the theorem is proved. 28. The determinant I '/ k, t, p t1,.... qitlk-_, Sk,... it)l )-l (m + l)k, n + 1, (p + 1)(mn -I).................. (t + 1)(+ + 1),_................................................................................ (? - + k)k, n + k, (p + k)(n + k),.................. (u + k)(lm + k),_(m + })k, n + r, (p + r)(n... + r)............... ( + )( + is equal to the product of (k+l)(k + 2)...r and i| mk, n p, p......k- (m + l)k>, i + 1, (p + 1) (m + 1 ),... (q + 1) (m + l))k-i (1) (mn + c)kî, k + k, ( + ) ( + k)... (q + k) (m + k)k-i That is to say, it is independent of the r - k quantities s,... u. To prove this, apply to the determinant the operations of II. 5 (iv). Then in place of any element P in the jth row we must write zAi-'P. Then in the first column every element after the (k + l)th vanishes, in each of the others every element below the leading diagonal vanishes, while the element of the ith column which is in the leading diagonal is (i - 1). Hence if we expand the determinant by Laplace's theorem, according to minors of the first k columns, it reduces to (k+ 1)( + 2)... r imk, n, pm1... qmk).-1, 1, [po,, + (l + l)o]. k_20,, 2(m,+ 1),................................,....... O, 0, O,... 108 THEORY OF DETERMINANTS [CHAP. VIII. which proves the theorem. For the last determinant is the result of operating, as in ni. 5 (iv), on the determinant (1). The determinant (1) is known by Art. 27, and hence we know the value of the new determinant. 29. Next let us consider i ndk, n Zd, p' d+1... t.d+r-i(m + 1), (M n + 1) (I m + l)d, (p + 1) ( + l)+i......;........................................................... (m + r)k, (n + r) (m + r)d, (p + r) (L + r)d+...... where k has any value from d to d + r- 1 inclusive. Divide the rows by ma, (m? + l)d... ('m + r)d respectively, and multiply the columns by kk-a, 1, (d + 1)i, (d + 2)2... Then our determinant is equal to md (m,- l1)d (m + 2)d... (m2 + r)d kk-d (d + l)l (d + 2)2.. (d + r-1).(. multiplied by the determinant (m - d)k-d,, (t- d)... (m-d+ l)k-d, n +1, (p+l)(m -d + 1)1.......................................... o..... (mi-d +r)kd, n + r, (p + r) (m-d+ r)... which by the preceding articles is equal to (k - d + 1) (k - d + 2)... r (m - g - ) (( - d -p - 1) (m - - q - 2)... (2), being independent of the last (r - k d) of the quantities n, p... u. The determinant we started with is equal to the product of (1) and (2). 28-30] DETERMINANTS OF SPECIAL FORMS 109 30. In the determinant of the last article let n=p...=u=, k= d = 1 then, if we multiply both sides by 21', we obtain mi, mrf, )2...i 'Y, (m + 1)i, 3 (m + 1), 3 (m + 1).... 3 (m + l)i. (m + 2), 5(m+2), 5 ( + 2),... 5 (m+2),...,............................................................................ (m + r)1, (2r + 1) (m + 1r)1, (2r + 1) (m + r1),... (2r + 1) (mn + r~),. = 2rm (m + 1)... (m + ~r). Divide both sides by n (m + 1)... (m + r), and then multiply both sides by r!, thus j 1, 1, (m - 1),... (m- 1),._i 1, 3,:3m.,... 3 (m),._i = 2.4.6... 2r. 1, 5, 5 (m + 1),... 5 (m + 1)r,_ I........................................ Hence, changing m - 1 into m, if we write Ur = i 1, M, 2... n,., 1, (m + 1)i, (m + 1)2... (m + 1)., 1, (m + 2)i, (m~ + 2)a.... (n + 2),. we have by Wallis' theorem 7r Lim. (2r + ) t2,._-=, when r, and therefore the order of the determinant, is infinite. CHAPTER IX. ON CUBIC DETERMINANTS AND DETERMINANTS WITH MULTIPLE SUFFIXES. 1. JUST as when n2 elements are given we can arrange them in the form of a square, so when nW3 elements are given we can arrange them in the form of a cube. Then we can indicate the position of the elements by means of three suffixes. The elements will lie in three sets of parallel planes; supposing the cube containing the elements to stand on a table with one face towards us, we may for convenience call those planes parallel to the face on which the cube rests strata, those parallel to the face in front of us planes, and the perpendicular planes sections. 2. An element of such an array will be denoted by aijk, where the suffixes mean that it stands in the ith stratuim, jth plane, and kth section. The set of elements in the leading diagonal will be a,,111 222 * * * a,,,,, From this we can form a function analogous to a determinant, and hence called a cubic determinant, by the following.process. From the leading term a1lc,22... ann we form n! terms by writing for the series of third suffixes all possible permutations of 1, 2... n, giving to each of these terms a sign corresponding to the class of the permutation. Then from each of the terms so obtained we derive n! new terms by writing for the series of second suffixes all possible permutations of 1, 2... n, giving to each new term, relatively to the term from which it is derived, 1-4-] DETERMINANTS WITH MULTIPLE SUFFIXES 111 the sign corresponding to the class of the permutation. The sum of all these {n!}2 terms is called a cubic determinant, and is denoted by +_ a clla222 *... Cnl,, or by ak (i, j, 2,...n). 3. Just as an ordinary determinant can be represented as the product of n alternate numbers, so a cubic determinant can be represented as the product of n factors lineo-linear in two sets of alternate units. If el, e2... e2; e1, e2... en are t.wo independent sets of alternate units, then the determinant of Art. 2 is equal to the product II {ailieeld + ai2,e~e2 +... + ailne1 e + ai2162el + ai22e2e2 +... + ai221e2en 1, 2 (i = 1, 2... Tij......................................... + aim n el+ E ai2, 21,2 +.. + ainn2nen e For if we consider any term of the product, it will vanish if it contains two e's or two e's with the same suffix, i.e. if two a's with like second or third suffix occur in the term, which ensures that all terns which do not belong to the determinant vanish. Thus every termn which does not vanish contains some permutation of the units e1, e2... en and e, e2... en as a factor, and if the units be brought to this order the sign of the term will be (- y)L+"; where,u is the number of inversions in the e's, i.e. in the second suffixes of the term, and v the like number for the e's or third suffixes. That is to say each term of the product is a term of the determinant with its proper sign. Thus the determinant is correctly represented by the product. Just as an ordinary determinant is the. product of linear functions of the elements of a row, a cubic determinant is the product of linear functions of the elements of a stratum. By means of this representation we can deduce the properties of cubic determinants. 4. The sign of the determinant is changed if we interchange two planes or sections. For interchanging two planes is the same thing as inter 112 THEORY OF DETERMINANTS [CHAP. IX. changing two e's, and interchanging two sections the same as interchanging two e's. Either of these changes alters the sign of every term, and therefore of the whole determinant. 5. Interchanging two strata does not alter the sign of the determinant. For we can represent the determinant by either of the two products I (be + be, +... + b,,e ) (i = 1 2... ), ^ i - 1, +.+. b i l II (Cilel + Ci2e2+. + Cin En) where bik = ailk e + ai2ke2 +... + aink en Cik = aikle + aik2e2 +... + (aikn enl. From the first form we see that the determinant, on interchanging two strata, suffers a change of sign as being the product of alternate numbers belonging to the system e; from the second we see that it also suffers independently a change of sign as being the product of alternate numbers belonging to the system e. Thus on interchanging two strata the determinant undergoes two changes of sign, and hence remains unaltered. 6. A cubic determinant of order n is the sumn of n! ordinary determinants, each of order n. For as in Art. 5 A = II (cilel + ci2).. +.+ Cin) where cik has the same meaning as in Art. 5. Hence, by I. 19, A = Cik. Thus the cubic determinant is equal to an ordinary determinant of the same order, whose elements are alternate numbers. To split up this determinant into others with simple elements we must take a partial column from each column of the determinant, but if we take a partial column in the pth place from one column we cannot take a partial colurnn in the pth place from any other column, for then ep would occur twice, and the corresponding determinant must vanish. Hence each selection of partial columns must be a permutation of 1, 2... n, there are ni such selections, and as many determinants with simple elements. 4-8] DETERMINANTS WITH MULTIPLE SUFFIXES 113 Thus A = laij(k)I, where the determinant on the right is an ordinary determinant; k is put in brackets to remind us that though it varies from one column to another, in the same determinant it remains fixed. This theorem is also an obvious consequence of Art. 2. 7. If in the preceding article we suppose all the first suffixes to be the same, all the determinants on the right would become alike, only their columns being permuted, and each determinant would have the sign corresponding to that permutation, hence suppressing the first suffixes altogether, the cubic determinant is now equal to (n!)a jkl (j,- k l,2...n). This then is the value of the cubic determinant whose strata consist of the determinant al... ain ani... ann repeated n times. 8. The product of two ordinary determinants, each of order n, may be expressed as a cubic determinant of order n. Let A=-aikl = A A2... An, B= Ibikl= BB2... Bn, where Ai = aile, + ai2eS +... + ainn), Bi = bile, + bi2e2 +... + bine, the systems of units e and e being independent. Then AB= nrAiBi = II (ailbile el + ailbi2 e e2-... + + aibine en + ais2b e2el + ai2bi2e2e +... + ai2bin e2e.......................................... + ain bilenel- in bi2 n e +... + ai in en en). Now if Cijk = aijbik, the product on the right is the cubic determinant of the elements Cijk. Thus the theorem is proved. S. D. 8 114 THEORY OF DETERMINANTS [CHAP. IX. By multiplying Ai and Bi together we avoided any inversion of the A's and B's among themselves. If we allow for the consequent changes of sign we can have as many such inversions as we please, and so vary the form of the cubic determinant which represents the product. 9. The product of a cubic determinant A, whose elements are aijk, and of an ordinary determinant B, whose elements are bik, is expressible as a cubic determinant C, whose elements are cijk, where Cijk = bjaitki + bj2a1f2 + *. + bjnaiki, To prove this, we observe that C = II (cilleel + ci12ee12 +... + cinee,, + c21e el + ci22e22 +... + c442 e 2ez +...................................... + Cini nei + Cin2 ene2 +... + Cin n en) = II (ai, B1 el + ai,2B e2 +... + ai1n BI e2 + aiB2el + aci22B2e2 +... +.......................................); where Bj = bye + b2je2 +... + bnjen. Now since the alternate numbers Bj follow the same laws as units, this last product is a representation of the cubic determinant A by means of the units e and B. Thus C =A. el... e,. B... Bn = AB. 10. It is now an obvious step to construct functions analogous to determinants by means of letters with more than three suffixes, though when we take elements with more than three suffixes we cease to be able to picture to ourselves their arrangement topographically as we can in the case of elements with one, two or three suffixes. We can, however, conceive a set of elements with p suffixes such as aijk... 1, np in number, to be arranged in p sets of rectangular planes in a space of p dimensions, and forming a rectangular parallelo 8-12] DETERMINANTS WITH MULTIPLE SUFFIXES 115 scheme of p dimensions. (Cf. Schlafli, Quzarterly Jour. II. p. 278.) The elements which have all suffixes the sanie, except i, lie in the same line, those which have all suffixes the same, with the exception of i and j, lie in the same plane,... those which have only i in common lie in a rectangular paralleloscheme of p- 1 dimensions. The product of the elements all...1 a22...2 ~~' a~n... ^ is called the leading term of the determinant of the pth class, which is formed by keeping the first suffixes unaltered, and writing for each set of the other suffixes all possible permutations of 1, 2... n. To each term so obtained we give the sign corresponding to the sum of the number of inversions in the p - 1 sets of variable suffixes. The whole number of terms is {n!}~-1. 11. The determinant of the pth class can be represented as a product of linear factors of the elements which lie in the same paralleloscheme of p - 1 dimensions. If e1, e2... e, c1, E2... ~q. 'qi, q12.. ait be p- 1 sets of alternate units; it is plain from reasoning similar to that in Art. 3, that the function A= II;,aij.ejek... e l (where the sum is formed by giving to each of the suffixes j, k... I all values from 1 to n, and then forming the product of such sums for the values 1, 2... n of i) is a determinant of the pth class and nth order, such as we have defined in Art. 10. 12. This definition is strictly analogous to those for determinants of the second and third class. A determinant of the second class is the product of linear functions of the elements of a row, one of the third class the product of n factors linear in the elements of a stratum. Here the determinant of the pth class is the product of n factors linear in the elements of a parallelo-:scheme of p- 1 dimensions. 8-2 116 THEORY OF DETERMINANTS [CHAP. IX. 13. It is clear that by the interchange of any two suffixes, except the first, the determinant changes sign. Also since the factors of the determinant can be written as linear expressions of each of the p -1 sets of alternate units, it follows that by the interchange of two first suffixes the determinant undergoes p- 1 independent changes of sign. Thus the determinant remains unaltered or changes sign according as its class is odd or even. 14. We have kept the first suffixes in their natural order. It is however indifferent which set of suffixes is retained fixed. If the class of the determinant is odd, it is perhaps more symmetrical to keep the middle suffix unaltered; the determinant is however not the same as before. 15. The product of a cubic determinant A, whose elements are aijk, and of an ordinary determinant B, whose elements are bik, can be represented as a determinant of the fourth class C, whose elements cijk are given by Cijkl = aijkbil~ For A = II (ailnelel + ail2e1e2 +... + ailmele + ai21 e2el +.................. ), B = II (bilq + bi2.2 +... + biln). Thus clearly AB = I (ijklejek) In = 1, 2 (In 1 i = 1, 2... n) which proves the theorem. 16. The product of two cubic determinants A and B, whose elements are aijk and bijk, both of order n, can be represented either as a determinant of the fifth class, whose elements are Cipqrs = Cipq bits) or as a determinant of the fourth class, whose elements are given by Cijkl = apjbpkl (p = 1, 2... n); the order of both determinants being n. The first part of the theorem is proved as follows: A = Hnaipqepeq. (In p, q=l, 2...n; in H i=1, 2....) 13-17] DETERMINANTS WITH MULTIPLE SUFFIXES 117 B = -IIHb.sj.ks. (In Z r, s=1, 2...n; in II i=l, 2...n.) Thus AB = Iaipqbis p, eqjr ks = IIZcipqrsepeqjrks. (In p, q, r, s = 1, 2... n; in II i= 1, 2...n.) Which by definition proves the theorem. For the second part of the theorem we have C = Il CiJkejek1l. Now the sum under the product sign = ej I{aij, + aj + a +... + ajjB} (), where Bp = bpllel1ii + b1,pl2el2 +... + bpq1b,1Te1 + bp21 e2 rl + b122e2aj +... + bp22,e2zn and if we write Aiq = aiCq e + ai2q e2 +...+ citq en the sum becomes BiAil + B2Ai2 +... + BAi,. The product of this has to be taken for all values of i. It must always be taken so that in each term we have the product B1B2... B,,; for if two B's are repeated the term vanishes. The value of this product is B. The remaining factors in the term are A1_pA2q... A,., where p, q... r is a permutation of 1, 2... n. This is an ordinary determinant of class 2. Comparing this with Art. 6, we see that it is a term in the expansion of the cubic determinant A as a sum of determinants of class 2. All these terms occur in our product. Thus C=A. B. 17. The following theorem regarding the product of two determinants of any class can be proved by the preceding methods. 118 THEORY 0F DETERMINANTS [CHAP. IX. The product of two deteriniants of classes p and q, whose elements are aij... and bij.. k respectively, can be represented either as a determinant of class p + q - 1, whose elements are Cij...luv...s -= aij...l bi u...S or as a determinant of class p + q - 2, whose elements are Cj...Iuv...s = s aj..., biu s (z = 1, 2... s), all the determinants being of order n. 18. It is not difficult to see how the theorems with regard to determinants of the second class (i.e. ordinary determinants) can be extended to determinants of any other class. It is probable that determinants of higher class possess many properties peculiar to themselves, though as yet not inany of these have been investigated. The complement of any element of a determinant is a determinant of the same class and next lower order. The extension of Laplace's theorem would shew how a determinant of class p and order n could be expanded in a series of products of pairs of determinants of class p and orders m and n - qt. 19. There is no difficulty in writing down the expansions of determinants of any required class or order. The number of terms however increases very rapidly. The following are the expansions of determinants of the second order, and classes 3 and 4 respectively: ~ (111)(22) = (111) (222)-(121)(212)+(122)(211)-(112)(221) + ~(1111) (2222)= (1111) (2222) - (1112) (2221) + (1212) (2121) - (1211) (122) + (1122) (2211) - (1121) (2212) + (1221) (2112) - (1222) (2111), while for the determinant of class 3 and order 3, ~ (111) (222) (333) (111) (()(222) (333) - (121) (212) (333) - (1) (232) (323) + (131) (212) (323) + (121) (232) (313) - (131) (222) (313) - (112) (221) (333) + (122) (211) (333) + (112) (231) (323) - (132) (211) (323) - (122) (2:31) (313) + (132) (221) (313) - (111) (223) (332) + (121) (213) (332) + (111)(233)(32) - (131) (213) (322) - (121) (233) (312) + (131) (223) (312) + (113) (221) (332) - (123) (211) (332) 17-20] DETERMINANTS WITH MULTIPLE SUFFIXES 119 - (113) (231) (322) + (133) (211) (322) + (123) (231) (312) - (133) (221) (312) + (112) (223) (331) - (122) (213) (331) - (112) (233) (321) + (132) (213) (321) + (122) (233) (311) - (132) (223) (311) - (113) (222) (331) + (123) (212) (331) + (113) (232) (321) - (13(21) (321) - (123) (232) (311) + (133) (222) (31 1). 20. We shall conclude this chapter with the following general theorems. A determinant of any class, all of whose elements are equal to a, except those in the leading diagonal which are equal to x, is equal to {x + (n - 1) a} (x - a)"-', n being the order of the determinant. We shall prove this for a cubic determinant, but the method is perfectly general. D = I (aele1 + ael e2 +... + ae el + aee, +... +... - xeiei...) = n {aEE' + (x - a) ei, where E = e + e + e,... + e,,, E' =e + e +.... + e,. Hence, since E and E' are alternate numbers, any term in which they occur more than once vanishes. Hence D = (x - a)" + a (x - a)"- E {EE'lekek} (k= 1, 2... i- 1, i+ 1... n);.'. D = (X - a)"L + na (x - a)n-1 = {x + (n - 1) a} (x - a)n-1; for Eel... e-ei+l... e e;... ei_ ei+l... en = (- 1)- ele2... e,,; and so E'e^... -ei1i+1... e, = (- l)i-1ele,... e,~. The last theorem of IV. 25 can also be extended to determinants of higher class. For a cubic determinant we may state it as follows: If all the elements in the ith stratum are equal to ai, with the exception of that which lies in the leading diagonal, whose value is xi, then the value of the determinant is f + a,f' (,.) with the notation given in iv. 25. CHAPTER X. DETERMINANTS 0F INFINITE ORDER. 1. IF in the symbol aik we suppose the suffixes i, k to assume independently all positive and negative integral values, including zero, we obtain a doubly infinite system of elements, which may be arranged according to the scheme * * * -2, -2 a-2, -1 o_-, o C-2, 1, 2 * * - ~ ~ * ( —, —2 ( —1,- [-1,O — 1,1 a-l,2 ~* ~...(,-2 a a,0, -1, a, 1 ao0,2..... —2 al, —1 al,0 al, al,2 *.. a 2, -.2 a2, -1 a2,0 a2,1 a2,2 * Thus if we take a pair of rectangular axes, and measure ordinates positively from left to right, and abscisso positively from above downwards, the position of aik will be at the point (i, k). 2. Let p, q be positive integers, and suppose that p + q=m; then if i and k each range from -p to q the determinant jaikI is of the rnth order and its elements are in the same relative positions as in the infinite array above indicated. Let m increase indefinitely in such a way that p, q both become indefinitely large: then the ultimate behaviour of laikl is analogous to that of an infinite series. It may become infinite; it may be indeterminate; it may converge. The case with which we shall deal almost exclusively is that iin which the determinant converges to a definite limit, A, which is independent of the way in which p, q become infinite. We may then, for simplicity, 1-4] DETERMINANTS OF INFINITE ORDER 121 suppose that i, k each range from -n to +n, where n is an integer which ultimately increases without limit. Thus A is the limit of the sequence Ao, A, A2,... An, *... where Ao = a0,0 and An = ak (i,k =-n...+) The element a0,0 may be called the central element, and a... 1,-, a0,0, a1,1,,,, — 1 a0, 0a l, 1 ' '~' the diagonal elements of the infinite determinant A. Any diagonal element may be taken as the central element; because if the notation is changed by writing ai+x,k+X = bi,k where X is any fixed positive or negative integer, the sequence which in the new notation is B0, B1, B,, etc. is simply one of the sequences which may be chosen to specify A. 3. If, in A, any two rows or columns at a finite distance from the centre are interchanged, the value of the new determinant is -A. For if we take n so large that A,, includes both the lines which are interchanged, the sequence An, An+l, A,+2,... becomes -An, -An+1, -An+a,... the limit of which is - A. In the same rnanner it can be proved that if two rows or columns are identical or proportional, A= 0; that if all the elements of a row or column are multiplied by k, the value of the new determinant is kA; that columns and rows may be interchanged, keeping the diagonal elements in their places; and so on. 4. The system of duads (i, k) may be associated with another system of duads (X, /), in which X, / independently assume the positive integral values 1, 2, 3, etc. The simplest way of doing this is to put X =2i if i is positive, = - 2i + 1 if i is zero or negative 122 THEORY OF DETERMINANTS [CHAP. X. and in like manner for / with respect to k. This correspondence is reversible; namely, if X is even, the corresponding value of i is ~X, and if X is odd, i =- (X - 1); with a similar rule for 1 and k. If, now, we write aik = b6, we can form a new array bil bl2 b13... bl b22 b23... b31 b32 b33 and derive from this a sequence BI, B2,... B,... where B, = Ib,. This is convergent, and its limit is A; in fact, B2n+- = A1 and Bn is a first minor of A,, which converges to the same limit as An itself. Accordingly it is sufficient to consider infinite determinants associated with an array of the second type indicated. It will be convenient to write to express that A is the value of the infinite determinant which is the limit of lnnl when the positive integer n increases without limit. 5. An infinite determinant is said to be normal if the product of the diagonal elements is absolutely convergent, and the sum of all the other elements is absolutely convergent. With the help of Kronecker's symbol Sik, these conditions are expressed by the single enunciation that the double sum 2 (ak - Îik) i 1 is absolutely convergent. Every normal determinant is convergent. To prove this, we write aik - bik == bik, and denote the absolute value of bik by /3ik. Putting - i=n c=n Pn= n {1+ 1ik}, i=1 c=1 4-6] DETERMINANTS OF INFINITE ORDER 123 the product P,n is absolutely convergent when n increases without limit, and Lt (Pn+p- Pn) = O for all positive integral values of p. If, now, A2 = ci \ and A,+p is the analogous determinant with p more rows and columns, the difference (A,+p -A,) can be expanded in terms of the quantities bik. Changing each term into its absolute value and prefixing the positive sign, we obtain part of the expansion of (Pn+ - P,,). Consequently mod (A,+p - A2) < P,+,- P,, and its linit is zero when n = ci. It follows that Ica] is convergent. 6. In a normal determinant A let the element aik be replaced by unity and all the other elements of the ith row and kth column by zero. The result is a normal determinant which may be denoted by Aik and called a first minor of A. If we put (- )i+kA ik =ik, we have k=oo A = a ia, + aii +...Y= E aik2k, and A = alkaCk + a2kcatk -+...= ti ck.ik. i=1 The truth of these formulae is almost self-evident. Let us take the auxiliary determinant ian,,, where n >i; then aCnn = aiLC il - ai2 ad2 +... + aCiini, i, where aiâ, etc. are first minors. Hence k=oo k=n k=oo 2 aikaOik -a,,tl = aik (aik - ik) + iZ aiklik. k=1 k=l k=s+l Now let n increase indefinitely: then the first sum on the right vanishes because each factor (<ik - aik) becomes infinitesimal k=n and Z aik converges; while the second sun vanishes because k=1 124 THEORY OF DETERMINANTS [CHAP. X. the sequence aik (k = n 1, n + 2,...) has an upper limit, and k=oo 2 aik ultimately vanishes. Therefore kc=n+l k=00 atikaik = Lt lan, = A; k=l n=oo and the other formula may be proved in a similar manner. 7. The notion of a minor of a normal determinant may be extended as follows. Let Pi,, P.. Pr qi, q2,... q1' be any two sets of positive integers, those in each set being all different. In the pith row and qith column of the normal determinant la,, replace the element cp.q. by unity and every other element by zero. After doing this for the values 1, 2,... r of i the original determinant has been transformed into another, likewise normal, which we shall denote by (, p2... pr \qi q9... qr This is an rth minor of \a,] and is complementary to the finite minor lapgij (i, —j=1,2,...r). If in the determinant la,,l we simply omit the rows and columns specified by p,... p. and q1... q,. respectively the new determinant is the minor above defined multiplied by the factor (- l), where / = E (pi + qi). Iff, g... 1 is a permutation of 1, 2,... r containing X inversions Pf Pg ' pl (-1) IPl P2 '" r qf qg.. qi ql q2. q, 8. From the first two rows of a normal determinant laOl we can derive an aggregate of determinants ahk defined by ialh alk a2h a2k where h, k are any two positive integers such that h < k. These 6-8] DETERMINANTS OF INFINITE ORDER 125 finite minors of the second order may be arranged as follows ini a linear progression: a12 a13 a14 a23 a15 a24 a16... the rule being that ahk precedes a, if h + k < 1 + m, or if h + k = I + m and h < 1. With this notation la ~hk ( (h =,,...), the terms on the right being arranged according to the rule just explained. To prove this we take the expansion lann = a.rsA s (r s = 1,2,n.n) = Si + S2, where Si = SlhkAhk (h + k < n + 1), S2 = \nnl - S, and consider what happens when n increases indefinitely. Ultimately the typical term of S, converges to akh, ( )j and since all values of h, k are included for which h + k < n + Si itself converges to Sah, (h, k = i, 2, 3,...) as above defined, provided that, before going to the limit, the terms of Si are properly arranged. The limit of Ss is zero, because if we write it S2 = alnAthe absolute values of the quantities AIn have an upper limit, and 2al. ultimately vanishes. Finally, lann converges to A, the value of 1 cl In a similar way, by selecting minors from the first two columns of A, we obtain the expansion I!I,,i a ak 2 hi (, k\, a..) aki ak2 1, 2)' 126 THEORY OF DETERMINANTS [CHAP. X. 9. In a similar way it may be proved that if pl, p,...P,. is any fixed selection of r different whole numbers, arranged in ascending order, A = a/o = Za(vq) (, p ) q \ql, q-2, q, where on the right hand a(pq) denotes the finite minor =((Q) Cpqj (i, j = 1, 2,... r), and qi, q2,... q is any selection of r different whole numbers arranged in ascending order. It must be borne in mind that A is normal, and that the sets (gq, q2,... q) must be arranged in linear order by a suitable rule. The summation includes all the selections ql, q,... q,.. There is a corresponding expansion with qi, q2,... q. a fixed selection, and pli., p,... p a variable selection. 10. The product of two normal determinants A, B may be expressed as a normal determinant by a method precisely analogous to that used for finite determinants. Thus, if A = la,,, B=\b^l, = =cO,|, where Cik = aibk + ai2bk2 +.. = aisbks ( = 1, 2, 3,...), then C is a normal determinant and its value is AB. To prove that C is normal, write aik = ik + alik, bik = ik + b'ik, Cik = Îik + C'ik; then CGik = Caik + b ik + a'isb ks, and therefore |C Cik | ai |,k + S b + a' b'is | (iI, s= 1,2,3,..). Hence, A, B being normal, Z C'ik is convergent, and therefore C is normal. Again, AB = Lt ca,,,l b,,, = Lt |,,!, where 7ik = aiibk + aiabk2 +- *. ainbkn (i, k = 1, 2, 3,... n), and it can be proved that Lt Iyi, = C. 9-11] DETERMINANTS OF INFINITE ORDER 127 To do this, write ^ik =ai, + bk,f + f i, n+2 bk,l +2 +.. -= aitbkt (t =? +- 1, 2n + 2,...), t then cnl?= | nn + rnnl = 1 + r C11 Ci+ i112 C2 +... + rln, Cl, where Cnl, C1,... Cmi are first minors of cnL. Now these first minors are all finite determinants, and a positive quantity R can be assigned which is greater than the greatest of their absolute values. Consequently, if pik denotes the absolute value Of Tik, mod |Cnn - |7n11 R (l + p2 +...* * * + pl), which ultimately vanishes when n = o. Therefore Lt ICnnl = Lt 7ml| = Lt \ann\ Ibnn\ = AB, n=oo n=oo which proves the theorem. As in the case of finite determinants the product AB may be constructed in four different ways (v. 4). 11. There is a class of infinite determinants, which we shall call semi-nornal, defined as follows. Let il, X X2 3, ~ ~ lYI, Y2, Y3,.. be a series of quantities such that Pi -= cnI 'O = = - i yr is absolutely convergent. Suppose also that A= [a,O,1, B= | ^ib where bi = - ai. Then if A is normal, B is also normal: but it may happen that B is normal when A is iiot. In this case A is said to be semi-normal, and the system (x., y,.) may be called a reducent of A. Clearly if one reducent exists, there will be any number of them. Since =bjnPi = Pn lat\, where P,, is the product of the first n factors of P,, and since 128 THEORY OF DETERMINANTS [CHAP. X. when n is infinite b,,n and P, converge to the limits B and P, it follows that A is convergent, and that A = P,-B. 12. Under certain conditions the product of two semi-normals may be expressed as a semi-normal. Let A=la,6l, B=\b,,l be two semi-normals with the respective reducents X ^ X2 XS... SjZ Z2 Z3... yi y2 Y3 y. (.U1 u 2 3... then if the product Pw = II 1 Za, is absolutely convergent, AB = C= Ic,, where Cik = aai1bhk (h = 1, 2, 3,...), and C is a semi-normal determinant of which Xi Xs2... U1 U.2 U3... is a reducent. To prove this, let us write Xi Zi aik = -i, k = -t bik, Yik = aih3hk; Yk Uk 1 then Cik = - Oih hk = hk - i aihhk, Xi Zh Xi Zh and therefore mod Cik < ju mod ( Yik) where / is the upper limit of the quantities mnod (yh/zh). This upper limit exists, and is finite, because r1 y h h Y h 1 Zh Xh '6h Zî all the products on the right-hand being absolutely convergent. Thus the series Cik is absolutely convergent. Again if we put aik = -ik + a ik, 3ik = 3ik -+ /3ik, 11-12] DETERMINANTS OF INFINITE ORDER 129 where 8ik, as usual, is Kronecker's symbol, Cii = -U ( + + ii i + y i -a ih hi), xizi h yi Zh and hence the product Icii is absolutely convergent, and so also xi is H - Cii. 'Ui Finally the series Xi Cik= z-h rih3hk i, kI k Zh is absolutely convergent, because 'aih/3hk iS so and yh/zh has a finite upper limit. The theorem stated has therefore been proved. In the same way it can be shewn that if IxhZh is absolutely convergent 1awOwol \b~ l = dcolJ) where Idol is a semi-normal with elements defined by dik = 2aihbkh, h and a reducent XX1, X2, X3 {\U-1, U2-l, 3... Moreover, in these enunciations, x and y can be interchanged, and also z and u. S. D. 9 CHAPTER XI. APPLICATIONS TO THE THEORY OF EQUATIONS AND OF ELIMINATION. 1. THE solution of a system of linear equations has already been partially considered (p. 26); we shall now proceed to discuss the general problem. Let us take the m homogeneous equations anx al + a x- +... + axlnx = a21 x + a22x2 +... + a2nxn =; amxxe +- amm22 +... + -annn = o the question is to find all the values of the unknown quantities xi which satisfy these relations. The nature of the solution is essentially connected with the matrix (an), which we shall denote by A. It follows from the partial investigation above referred to that if n = m + 1 and A is of rank m, the ratios of xi, x2,... xn are determinate: in fact X: x2:... n- A:~ A1:...: An, where AI, A,,... An are determinants of the nth order derived from A by suppressing one column. If, however, the rank of A is less than m, the determinants Ai all vanish, and the values of x,: 2...: xn apparently become indeterminate: the process of p. 26 is in fact illegitimate, because the derived system is not equivalent to the given one. We shall see presently that if r is the rank of A, and m < n, the complete solution will involve n- r independent parameters, while if m > n, the only solution is x = x2=... = x, = 0 unless r < n, in which case there is a solution involving n- r parameters. 1-3] EQUATIONS 131 2. Suppose, in the first place, that m = n, and that lannl is not zero. Let the reciprocal of lann be a,,,, and write Ui = a6ik-x, ( = 1, 2,... n). Then we have identically 2ÎikUi = isCikXs =-a,,, Xi k i,k and hence lannl i = O, or, since lannl does not vanish, X= 2... -= X = O. This, then, is the only solution when lannl is different from zero. Conversely if the equations ui = O can be satisfied by values of x1, x2,... xn which are not all zero, the determinant lanni must vanish. This determinant is called the resultant (or eliminant) of the n equations ui = 0: or again the determinant of the n linear forms ui. 3. Next suppose that m < n. Without loss of generality we may assume that larri does not vanish: let this determinant be called Ar and let its reciprocal be larrl. Then as in last article s=n aCliu+ o2iu2+... rir+ Ci, (=,... + =A + (= 1...r), s=r+l where Ci,s is a minor of A of order r, which may or nmay not vanish. Hence instead of the system u1 =u2 =... = u, = O we have the derived systein Ari = CisX, (i 1, 2,... r), r+l which is equivalent to it because arr = AAr-, which is different from zero. Hence we obtain x,, x2,... x, as definite linear functions of the (n - r) quantities xr+, xr+,... x,: it may of course happen that some, or even all, of the coefficients Ci, vanish. In any case, we have found a complete solution of ul = O, u2 = 0,.. - r = O, with (n-r) independent parameters xr+,... x,. Now let A,+1 9-2 132 THEORY OF DETERMINANTS [CHAP. XI. be the vanishing determinant jar.+l,r+ and i/3 +,.+ its reciprocal in which r+,r+-1 = A,., a quantity different from zero. Then we have an identity /3i,r,+l 1, +...,r+ + 3.,+,+ iUr+, +i = 0, because if the expression on the left is arranged as a linear function of x1, x2,... x, every coefficient vanishes as being a minor of A which is of order (r + 1). Hence every solution of the system U1 = u =... = Ur = O will also satisfy u,.+- =: and in the same way it will satisfy s = 0 for s= r+2, r+3,... m. Thus we have actually obtained the most general solution of the given system of m equations in a form which leaves (n - r) of the unknown quantities arbitrary. It may happen that some one or more of the quantities x1,... x, is definitely zero. For instance, the solution of x + y- 2z = O, 2x + y- 2 = 0, is x = O, y = 2X, z = X, with X arbitrary. 4. Next let m > n, so that the given system is redundant. Then if r = n the only solution is xi = 0, because without loss of generality we may suppose that lann is different from zero, and then the first n equations give xi = 0. If r < n, we may suppose that larr is not zero, and proceed as in last article: we thus again find a general solution in which (n - r) of the unknown quantities remain arbitrary. 5. Having given m linear homogeneous functions ui of n independent variables x1, x2,... xn, i is important to know how many independent identities exist of the form L (u) = X\u, + X2 U2 +... + Xmum = 0, with constant coefficients Xi. Equating to zero the coefficients of x1, x2,... xn on the left hand we have a1X 1 + a...X +... + amXm = O, al2X, + a22X2 +... + am2Xm = O,....... +... +..................... alX + a2nX2 +... + ar.nr = O. 3-7] EQUATIONS 133 This is a system with the matrix (amn)', the rank of which is the same as that of (an). Consequently if r is not less than the smaller of the numbers m, n, the system will have only a zero solution and the forms ui will be independent: while if r is below this limit there will be (nm-r) linear relations Li(u)=O from which all others can be derived in the form -CiLi (u) =, (i = 1, 2,... m- r), with arbitrary coefficients Ci. 6. A non-homogeneous system Ui + Ci = 0, (i = 1, 2,... m), where ci is a constant, may be reduced to a homogeneous one by putting xk= -, (-l,2,... n); yn+i it should be noticed that for particular values of the arbitrary parameters contained in the solution xi may become infinite or indeterininate. 7. Suppose that we have two sets of variables xl, x2,... x and yi, y2,... yn connected by the relations yi = Caikx, (i, k = 1, 2,... k the coefficients aik being constant, and lanl different from zero. From these equations we can deduce an equivalent set xi = - CikYk. k If we substitute in the first set of equations the value of xi given by the second set we obtain n linear equations in yi, y2,... y which rnust be identities, if, as we suppose, the variables Xl ) 2,... Xn are independent. Hence aikakj = Sij, k and in the same way by substituting from the first set of equations in the second t ikakj k= &j 134 THEORY OF DETERMINANTS [CHAP. XI. Consequently (ann) (n7n,) = (Oci) (afin) = [1], | = 1. If we denote the matrix (ann) by A, and write E for the matrix [1], then the matrix (ann) is conveniently denoted by A-1: thus AA-1= A-1A =E. This notation is consistent with the ordinary laws of indices; and we may express the relation between the sets (x1, x2,... x,) and (yi, y2, *.. yn) in either of the symbolical forms (y)=A (X), (x)=A-1 (y). These relations constitute what is called a linear substitution; thus in analytical geometry when we change from one set of coordinates to another of the same type, this is effected by means of a linear transformation. 8. If there are three sets of n variables (x), (y), (z) such that (y)=A(x), (z)=B(y), then it is found by direct elimination of Yi, y2,... yn that (z) = BA (x), x = (BA)- (z) = A-'B-1 (z), the products BA, A-lB-1 being defined as in v. 2 (p. 50). This theorem may obviously be generalised. Let u (x) be the linear form defined by U () =: ~x~, (k = 1,.... n), then by the substitution (x) = A (y) this is converted into v (y) = wlkyk, where = elaik +:2a2k +... + 4 nank, so that (7) = A' (e), where A' is the conjugate of A (p. 49). Thus the simultaneous substitutions x) = A (y), () = A' (e) transform 2 tix into q7iyi. The variables i are said to be contragredient to the variables xi. Variables transformed by the same substitution are said to be cogredient. 7-10] EQUATIONS 135 9. Let there be n linear forms ui (x) defined by Ui (X)= 2,pikXk, k and let them be transformed into linear forms vi (y) by the substitution x = A (y). Then vi (y) = Zqikyk, where qik = pirark, and consequently Iqnn = Ipnn. ' ann\i. Thus the determinant of the system of forms reproduces itself multiplied by lanl, which is called the modulus of the transformation. This is what might have been expected: for if (pnnl=O the quantities ui(x) are not independent: and when this is so, the quantities vi (y) are not independent either, so that | = 0. Given any system of forms Fi (x1, x2,... x,) the substitution (x) =A(y) converts them into forms Gi(yi, y,... yn). If a function of the new coefficients is identically equal to the same function of the old coefficients multiplied by a power of the modulus of transformation, we have what is called an invariant of the system of forms. It has been proved, then, that the eliminant of a system of n linear forms in n variables is an invariant. It follows from Arts. 1-5 of this chapter, as well as from Vil. 8, 9, that the rank of \qnl is the same as that of IPnnl. 10. In Art. 2 we have the first example of the process of elimination; namely, we have found a condition, independent of the variables, which must hold if a certain given number of equations are to exist between these variables. When r homogeneous equations hold between r variable quantities, (or what is the same thing, r non-homogeneous equations between r - 1 quantities), it is always possible to establish an equation R = 0 between the coefficients of these equations alone. Then R is called the resultant or eliminant of the system of equations. 136 THEORY OF DETERMINANTS [CHAP. XI. When the equations are two in number the most direct process is Sylvester's dialytic method. Let the two equations be 0 - ao +- ax - a2x2 -+ area;.m.............. (1) 0 = bo + abx + b.x2 + + Jbx. If we multiply the first equation by 1, x, x2... x2-1 we get n- 1 new equations, and from the second by multiplying by 1, x, x2... xmnwe get m - 1 new equations, viz. we have now the system O = ao + alx + a2x2 +... 0 = aox+alx2+... 0 = a2 +... O = b+ bo x + b2X2 +... O= box+ blx2+... 0= bo2. +.. of m + n equations satisfied by the same values of x as the given equations (1) and linear and homogeneous in the m + n quantities 1, X, X2 r. X+n-1 Hence, by Art. 3, the determinant of the system must vanish, or.R = a, a, a...... 0, ao, ai...... a0...... ooooo.............. bo, b1, b2...... b0, b1..... bo...... the determinant being of order m + n. Since there are n rows of a's, and m of b's, the resultant is of order n in the coefficients of the first equation, and of order m in the coefficients of the second. 11. If the coefficients a,, am-_, am-2... bn, b,_- bn-2... are functions of y and z of degrees 0, 1, 2..., it can be proved that the resultant is of order mn in y and z. This will be the case if every term in R has the sum of the complements of the suffixes equal to mn. 10-12] EQUATIONS 137 If we change y and z into yt and zt respectively, the value of R is now R' = aotm, altm-, at-2... ao tm, a tm-.................,oo o. botn, blt~2-l, b2t-2... botn, blt,-1... Observe that the separate elements and therefore each term of R' is multiplied by a power of t equal to the complement of the suffix. Now, multiply the first n rows by tîl-l, t-2... t, 1, and the last m by tm-1, tm-2, t, 1. Then R' is multiplied by a power of t, whose exponent is m (m - 1) n(n-1) 2 2 But now the first column of R' divides by tm+l1-î, the second by tm+n-2, and so on. Thus R' -R is equal to a power of t whose exponent is (m + n) (m + n - 1) m (m-l) n(n-1l) 2 2 2 Thus every term in R' must divide by tmn, which proves the theorem. Functions, such that the sum of the suffixes, or of their complements, of the elements in each term is constant, are sometimes called isobaric, and the constant sum is called the weight. 12. We may consider the question in another way. If b (x) = bo + blx + b2x2 +... + bxn = bn (x - 1) (x -/32)... (' -/n)............... (1) is an equation whose roots are /3,, 32... 3,, the function f(x)- = =a, + ax + ax2 +.............(2) has n values corresponding to the different values of x given by (1). These n values are the roots of an equation of the nth degree, which we now proceed to find. Multiply the equations (1) and (2) 138 THEORY OF DETERMINANTS [CHAP. XI. by the same powers of x as in Art. 9, and we have the m + n equations 0 = ao - u + ax + a2 +... 0= (ao - u) x + ax' +... 0 = (o-u) 2 +.........o....o........o 0= bo + bx +b2x2+... 0= box+bX2+4... =..................... Eliminating between these the quantities:m+n-1... x 1, we get ao — u, ai, a2... =0, ao - u, al. a. - u.....o.......... bo, b1, b2. bo, b.................... an equation of the nth degree to find u, the roots of which are f (/1), f (/ ),... f (/,n). The product of the roots being equal to the constant term, (-1)2 bf (/l)f (i32)... J(n) = (- 1)" R, where R has the meaning in Art. 10. Thus R = b:f (/)f/ ( 32)... (3). In the same way we may shew that R = (- 1)m (az) f (ai) (a2)... (am) if al... an are the roots of (2). This result shews that the value of R obtained in Art. 10 does not involve any irrelevant factor; for clearly (c1) (C02)... (Cm) is the simplest rational symmetric function of both sets of roots which vanishes when the equations have a root in common. 12-14] EQUATIONS 139 13. If the two functions b and f of the preceding article are a function and its differential coefficient, then R is called the discriminant of the function, and its vanishing is the condition that the function should have equal roots. If f (x) = ao + ax + ax2 +...+ axn =an (G - a,) (X - 2).... (x- n) f (x)= ai + 2ax +... + nanx-l, R = -lf' (a)f' (a,)..f' (a,) = a,, 2aa, 3a... a,, 2a2... oo..oo..... - a, a, a,... a,, ai... oo..............oo, having n rows of the first, and n- 1 of the second kind. If we multiply the last row by n, and subtract it from the nth, this becomes... 0, -nao, -(n7- 1)al,...-a,_l, O. Thus the determinant reduces into the product of a, by a determinant of order 2n - 2, which we shall call A. Also f' (a,) = an (a - a2) (a - a)... (a - an) f' (a2) = (a2 - a) (a2) *.. ( - an) f (an) = (an - a) (a, -a)(a, -a )... a,; z(n-l).' (al)f' (a,)... / (a,) = (- 1) 2 an " (a,, a... a,,) where (al... a,) means the product of the squares of the differences of all the roots. Thus n(n-1) a=(-1) a2 -2'- (a,, a... a,). 14. The artifice used in eliminating x between two equations may sometimes be employed for the case of more equations than two, as in the following examples due to Cayley. Let x+y+z=O, x2=a, y=b, z2=c; 140 THEORY OF DETERMINANTS [CHAP. XI. multiply the first equation by 1, yz, zx, xy, and reduce by means of the other three, then we get x+y + z=O xyz + cy + bz = 0 xyz + cx + az = 0 xyz + bx + ay = 0, whence, eliminating xyz, x, y, z, we get., 1, 1, 1 =0. 1,., c, b 1, c,., a 1, b, a, Or if we multiply the equation by x, y, z, xyz, and eliminate 1,, yz, x, y, we get., a, b, c =0. a,., 1, 1 b, 1,., c, 1, 1,. Again, if we are given the equations x+y+z=0, x3=a, y3=b, 3=c, if we multiply the first equation by x, y, z, y2z2, z2x, x2y2, xyz, y2zx, z2xy, and reduce by the last three we can eliminate x2, ) 2 y, zx, x y y, xy yz2, yz2x2, zx2y2 between the resulting equations, giving 1,.,.,., 1, 1,,.,. =0. 1, *, i. 1, 1,,,.,.,., 1, 1, 1,.,.,... c,., a,.,.,.,., 1,. b,.,.,.,.,.,., 1.,.,.,a. a,.,.,., 1, 1.,.,.,., b,.,., 1.,.,.,.,., c, 1, 1,. Other forms of the resultant can also be obtained. 14-15] EQUATIONS 141 15. The resultant of two equations has been obtained in a compact form by Bézout. Let the equations be f = am + axm-l +... + a = 0, <k = boxn + bi6x-l1 +... + b = 0, and suppose nm > n. Write fo = a 0 +a, f = a + a,+, f2+= ax a... = aoxr+ ClrT-l+... +a 0= = bo, 1 =b' + b1, (j = box2+b bx +b2,... = boxr + blxr-l +...+ b, and form the combinations X_= xm-nf<. -. = r^-nfr (b,+lxn-r-1 +... + bn) -,. (a,+,m-r- +... + arn) =- crsX-1, (s = 1, 2, 3,... m), for r= 0, 1, 2... (n - 1). If f= O and < == have a common root the m equations X0=, x =0,... X,_= 0, n, x = o, =... xn-n- -lS = 0, can be simultaneously satisfied, and hence eliminating 1, X, 2,... m-1 dialytically Co, l C0,2, *... C,m bn-, n-1,... =n- 0. bo0 bl,... 0 =O. O, bo,... O......,...o..oo.......... O, O,... bn It is easily seen that the expression on the left is of the proper dimensions in the coefficients of f and <, and that it does not vanish identically: hence it is the resultant, free from extraneous factors. As an example, let m = 3, n = 2: then the resultant is aobl - al b, aob2 - a2bo, - a0bo aob - a2bo, aib2 - a2bl, - ab1. bo, b1, b2 142 THEORY OF DETERMINANTS [CHAP. XI. When m = n, the determinant is syminetrical, and each of its elements is expressible as a sum of quantities of the type (aibj - ajbi). If in this symmetrical determinant of order m, we put bn+l = b~l+2 =... = bm = 0, the resulting expression is the product of am-" by the resultant of f and q as they are given at the beginning of this article. 16. We shall now prove that if Bézout's determinant is of rank r, the polynomials f, b have a highest common factor H of degree (m - r). For the sake of a uniform notation, let 1_=,r2 = x0,... *,m-n = xm-~, *m-11+1 = XO) * m-nt+2 = Xl, * *.m = Xn-1 i = ikxk-1, ( = 1, 2... m), so that the resultant is Rm= lPmml Let XI, X2,... Xm be constants, t any whole number not greater than m: then if we write = Xl, 1 +1 + X22 +...* + X,, = Co + C1 x +... + Cm-lm-l, the conditions CM-1C-2 = Cm *= m.-t == = form a deficient system of linear equations in X, X2,... Xm. This system always has a solution in which the quantities Xi are not all zero (Art. 3). For such values the degree of S is less than m-t. Nowr can be expressed in the form Af- Bf where A, B are polynomials, so that * is divisible by H. The degree of H cannot exceed mn: let it be m - /-. Then by putting t = /, we infer that from the equations Cmi_1 Cm-2 =... = 0Cm- = the other equations Co = = C. =. C.. _ = O necessarily follow: in other words the system Co = Ci =... = Cm- = 0 does not contain more than /L independent equations. But the actual number of independent relations is precisely r: consequently,/ > r, and the degree of H cannot exceed m - r. 15-16] EQUATIONS 143 The general values (Xi, X2,... X\) which make T identically zero can be expressed as linear homogeneous functions of (m - r) arbitrary parameters. Now in the identity = Af-B the value of B is - (1 + Xx +-... + X.mXm-2) - *- (Xm-n+ifo + m-n+2 +... + * * mfn-) of which the degree is m - 1. Equating to zero the coefficients of Xm-i1 Xm-2... r+2 Xr+l, we get a system of (m - r - 1) linear equations which determine the ratios of the (m - r) parameters: substituting these in A and B we get an identity Af-B'= 0, in which the degree of B' does not exceed r. Since B'4 is divisible byf, it follows that f, p have a common divisor of degree not less than m - r. We have already seen that the degree of H cannot exceed m - r: therefore its degree is exactly m - r, as stated. In fact H =f/B' = 1/A'. A numerical example will illustrate the argument. Let f= x4 x+ 23 + x + 1, =3+ X3 + x2 + 1: then R= 1 1 1 1 i, 0-1 0-1 -1 -1 -1 -1 0-1 0-1 = =x 3 + +x+1, 3 = (24 + )_-( + l)/=- - x- - _= x4 -f= - 2 - 1, 4 = (3 +x 2+2x),-(.2+x+l)/=-_2-1. The identity Xif1 + X2fr2 + X3F3 + X4*4 = 0 requires that X - X\ = 0, xI - x2- - - 4x = 0, Xi - X3 = 0, X1 - X.2- X3 - X4 = 0, whence X,, X2, X3, X4 = t, u, t,- U, 144 THEORY OF DETERMINANTS [CHAP. XI. where t, u are arbitrary. This gives 0 = X i = [t (x2 + + 1) - u (x3 + x2 + x)] - [t ( + 1) - f (x2 + i)]. To reduce the degree of the polynomial which multiplies <, we must put u = 0: thus finally, (x+x + 1) -(x+ )1)/= identically, and H == /(x + 1) = x2 + 1. 17. The resultant of the quadric u =allx,2 +... + 2aikxiXk +... =............(1), and of the n - 1 linear equations V1 = Cllxl + C12x2 +... + Clnxn = (.........................................................(2 ) Vn-x1 = Cn-ii X + Cn-12X2 +x...* + Cl-lnx = O can be readily expressed as a determinant. By Euler's theorem for homogeneous functions we can write the first equation in the form du d'u du xI dx + x2 X +'+ Xn+ x = 2 = O............(3). Then if in equation (3) we do not consider the variables implicitly contained in the differential coefficients, (1) and (2) being n equations between x... x,, (3) must be capable of being put in the form X1v1 + \X22 + +...Xn-_ =...............(4). Equating coefficients in (3) and (4), axi + ai2x2 +...+ ainx = cXc + X2C21 +... + Xn-lCn-l a21 xi + a22X2 +... + anXn =X1 c12 + X2c22 +...+ Xn-1-12 + (5) amixi + an2X + - - ann n = Xl cln + X2 2n +... - Xn-lCn-lnJ The equations (5) together with (2) form a system of 2n - 1 equations between x1, x2... x,, X\, 2... X,1; hence their determinant must vanish. Thus 16-18] EQUATIONS 145 ll1... alim l. *. cn-1 - O, anm... ann> Ce i.* Cn-in C11 *.. Cln............... Cn-1ii ~ Cn-in the blank space being filled with zeros. This result is due to Versluijs. aik and aki mean the same thing, viz. half the coefficient of xixk in the quadric. 18. The system of equations x+y=a, x2+y2=b2, is solved by establishing the new linear equation - y = + /2b - a2. Following up this idea Baur has solved the non-homogeneous system of an n-ary quadric and n - 1 linear equations between the variables; viz. let the system be allxl +... + 2akx xk +... =...............(1), Cl1+ +... + CinXn = Yi î C2lX1 +... + C2nXn Y2 C +.................. (2 Cn-11X1 +... + Cn —nXn = yn-1 Then we wish to establish a new linear equation C xi +... + cnnXn =.............. (3), so that if we determine the values of xi... xn in terms of y,... yn from (2) and (3), and substitute their values in (1), the result shall only contain yn in the form yn2. We are to have then = y2 + bikyiyk ( = 1,2... n- 1).........(4). Now if C= I Cik we have Cxi = Cliyi + C2iy2 +... Ciyn...............( Hence, differentiating (4) partially with respect to yn, we get du dx, du dx, du dxn dx, dyn d dx dy dxn dy' S. D. 10 146 THEORY OF DETERMINANTS [CHAP. XI. or, by aid of (5), if i = d cd Cyn = U Cni + U2 Cn2 +.. + Un C CD11 C12 9 Cin C21, C22... C2n..........................................(6 ). l-1l ) Cn-12 * * Cil-in U1, U2... un Substituting for the differential coefficients their values we determine the form of the equation (3). We have still to determine the value of y,. To do this we introduce the n (n- 1) quantities el,, el2 -. e e1, e1 22... e1n e211, e02.. e,, 6n-ii -n-l2 *. * ~an —in? such that er.i ak + ea a2k + *-. + eGa, = Ck; and hence Ae,.i c,.lAil + c,2Ai2 +... + crlAn, where A = aik. Thus A el,, e =... cl, = Cy,,......(7). e,,-,,, en_12...~ e,,n-m C9i-ll '~ C1 —1l Xl1, X2 **.. Xn 1... *,"l Now from the product of (6) and (7) C2y2 = A e,,... e,. cl... c,........................ en-il... en-iI cn-l... C -1in =A B11, B12... B y,,_, 1 Yî, Yî......................... Bn-ll, B71-12* * B _n-ln-,i y..-i... ( 8, YI, Y2 ' ' y2-1, U 18-19] EQUATIONS 147 where B,.s = c,.esl + cç,es22 +... + cr.,nes,, AB,.s = crl (siAni + cs2A12 +...) + C,.2 (csi A2 + s2 A22 +...) _= - O, C.1, Cr.2.. Cm -= -ABs.. Csl, a11, a12.. aln..................... Csn, al, an2... ann On the right-hand side of (8) all the quantities are known from (1) and (2). Thus Cy, is known; substitute its value in the left of (6) and we have the required equation (3), which with the equations (2) forms a system of n linear equations sufficient to determine the quantities x... xn. 19. The equation a1, - X, al2, a1,... i,, 0 a,21, a22 - X, a23... a2, (tl, a 12, aa,... an -X (where aik = aki) formed by taking X from each of the leading elements of a symmetrical determinant is of considerable importance in analysis. The following proof that its roots are real, when the quantities aik are real, is due to Sylvester, If we denote the left-hand side of the equation by < (X) we have (- X) = CIl, + X, ai,... ain a21, a22 + X... a^2 aCL, a,,2... ann1 + X and hence b (X), (-X)=- C - X2, C2... C c21, C22 - *.. C2n *.......................... cra, ~c92... C1 -- Cml î C2n * * * Cnn - X where cs = a,.lasl + a,.,ca2 +... *+ a. asn,; 10-2 148 THEORY OF DETERMINANTS [CHAP. XI. the X disappears, because a,.s = a,.. Hence, expanding the righthand side by Art. 23 of Chap. iv., (X) C (- X) = C - 2 IG + X4C2 -... + (- X2)1. Now, by v. 9, C1, C(2... are all sums of squares, so that the coefficient of each power of X is positive. Hence, if we equate the right-hand side of this last equation to zero, Des Cartes' rule shews that it cannot have a negative root. Thus X cannot be of the form 3 Y/-1. In order to shew that it cannot have the form a -+ f /- 1 we have only to write a, - a = ai/', &c., and the case is reduced to the preceding. CHAPTER XII. RATIONAL FUNCTIONAL DETERMINANTS. 1. IF we have a series of n quantities x, y, z... z, t we shall denote the product of all the n (n- 1) differences, obtained by subtracting from each number all that follow it, by " (x, y, z... u, t). So that ~ (x, y, z..., t)=(x- ) (- )... ( - t) (y- z)... (y - t) ( - t). This function i (x, y, z... it, t) is an alternating function of all the quantities x, y, z... t; viz. on interchanging any two of these it changes its sign, but not its absolute magnitude. It is thus of the nature of a square root, having two values equal in absolute magnitude, but opposite in sign. This is conveniently indicated by the index ~. The product of the squares of the differences will be denoted by ~ (x, y, z... u, t), and is a symmetrical function. This notation is Sylvester's. 2. We have x-1, x-2, 1 =.. (, y, z... t). y-l yl-2..y, 1 tP-l, tn-2... t) 1 For the determinant on the left vanishes if any two of the quantities x, y... t become equal, because then two rows become identical. 150 THEORY OF DETERMINANTS [CHAP. XII. Thus the determinant divides by the difference between each pair of the letters, being a rational function. Hence it contains x (x, y... t) as a factor.. But the leading term in the determinant is -~ly~-2... u. 1, which is also a term in 2 (x... t) with its proper sign. Thus the theorem follows. 3. By a similar argument it may be proved that if a rational integral function of x... t changes sign when any two of the variables are interchanged, it is divisible by - (x... t). 4. If fi (x) be a function of the ith degree in x, the coefficient of whose highest term is unity, we have fn-1 (X), fL-2 (X)... fi (X), I = y (x, y... t). fn-l (y), f/-2 (y)... f (y), 1.................................. fn- (t), f- (t)... fi (t), 1 For if we subtract the last column, multiplied by a proper number, from the last but one, the elements in this column become x, y... t. Now multiply the last two columns by the proper numbers, and subtract their sum from the last column but two, the elements of that column now become x2, y... t2. By proceeding in this way we reduce the determinant to that in Art. 2. If the coefficients of the highest powers of x are not unity, the determinant is equal to ~ (x, y... t) multiplied by the product of the highest coefficients in the separate functions. For example, if fi() =- (x - )... ( -- ( +, xil-1, -2... wX, 1 _ _ (X, y... t) Yn-, yn-2... Y (n (n -2)!........................... tn-l, til-2... * tl, 1 The denominator can also be written 2n-2. 3-3... (n - 2)2. (n - 1). 2-6] RATIONAL FUNCTIONAL DETERMINANTS 151 5. If fi (x) = alix"'-l + a,2x-2 +... + ai, we see by the theorem for multiplying two determinants (v. 4) that fi (Xl),./2 (i).. JI f (ai) = all... al Xl-1, -2... 1 fl(x,), f,2( ).................................... a... am x* * n~-1x, ~Xî-2... 1 fi (xn), fa (n)... fn (XI) = |a, ' (X, 2... wn). If fi (k) = (Xk -i)an,, = 1, Ci (- yl), C, (- yl)2... yl)n-1 1, c, (- y2), C2 (- y2,)' *.. (- y)l C(( —yn), C2(-yn)2... (-y) = C (yi, y *.. Yn), where C is the product of all the binomial coefficients of order n-1. For the elements in each column of the determinant are multiplied by that power of- 1, which is introduced by moving the column from its place in t to the place it occupies. Thus (X1 - y,)n-i, (x, - y2)n-l... (,i -yn)?-1 (|2 - yJ1)n11, (.2 - ya)-1... (2 - yn)Y-1................................................ (X - yl)n-1, (x - y2)n-... (x - yn)-11 = c ' (x,, X... n) ~' (Y i, y2 *... yn)If xi = yi this gives us ' (X1... XI) in the form of a determinant. 6. We may give other determinant forms to the product ~ (x1, 2... xn) (Yi, y2... yn). Thus ~ (x, x2... ***,) ~ (y,, y.*... )= yn-1... -1 ' | X12.-i. 1 y n-i 1 =I Cnn I 152 THEORY OF DETERMINANTS [CHAP. XII. where, if we multiply by rows, Cik = X -1 yn-1 + x-2yk-2 +... + x yk + 1 (xiiyk) — i Xiyk- Or if we multiply by columns Cik = X ln-iy n-k x2 yn-iy2n-k +... I+,ln-i n-k If we put xi = yi and si-= x + x.. +..+ x we get ' (X1, 12... I) = S2nl-2_ S2nl-3... Sn-1 S2nf-3 S221 —4... 82-2..................... Sn-i1, S8-2... So - So, 8...1 |~8 -S1, S2... 58n-2 Sl5-1, S 1 S 2n-2 an orthosymmetrical determinant. 7. A more general theorem is the following. Consider the array m-i, Xm.. x1, i m —1 Xi- m-2, 1, 2 > t2 * 2ï- J x-1 XI m-2.., 1 where n is greater than m. By compounding it with its conjugate, we get a determinant of the mth order which is equal to the sum of the squares of the nm determinants, obtained by taking any nm different rows in the array. The determinant has for elements Cik x i-lx k-1... + i-1X?k= Si+k — Hence, by aid of Art. 6, we get {(XXp, Xq...)j == So, Si... Sn-i| S1, S2... 'S ~.......*........... Sm- ~, SM m ~ S 28 ---2 where xp, Xq... are any m of the n quantities $x, x,... x,. 6-9] RATIONAL FUNCTIONAL DETERMINANTS 153 8. We have clearly by Art. 2 x",,,"-'.., 1 = (a, a...)f(x),n, a,"'-... a,, 1 where f (x) = (. - a) (x -,)... ( - a,,) = x" - pl "-1 + pn- _ -... + (-1 )n-< pn-ix +... By equating coefficients of xi on both sides we get i ""+" -1... a... 1 (a... a)pn-i a nn... ani+l,,i-1.... where pi is the sum of the products n-i at a time, without repetition, of the quantities a,... a,,. 9. We may write the first identity of the preceding article in the fbrm a',", a,~... a,, O 0 (- l)n (a,, a,.. an)f(), a,1-1, at-1-l1... a1 —', 1yXI-', 0,............. 0........... 0.........0 1, 1... 1, 1, O 0, O... 0, 0, 1 and similarly Ct1, an... an,, O, y" = (' )... a,)f/(y). al1"-1, a2n-1... nl^-l, O, 0 y?-l al, a,... an, 0, y 1, 1... 1, 0, 1 O, O... O, 1, 0 Form the product of these two determinants by rows; thus Sm,, Sm~... Sn, ~n = - (a,, a,... al)f(x).f(y), Sn ~n-i SQ, 1n-i y~, y~1... 1, 0 1544 TIIEORY OF )DE'I'TERMINANTS [CHlAP. XII. froni which by cquatiig coefficients of the powers of x nild y wc get a number of theorclms. s,. is now tile sun of the rth powers of the roots of the equation f(x)= O. 10. We nmay extend the theorenl of Art. 8 as follows: the value of the determinant,ln+r-l, Xn+r-2..., 1 gIt' ) w\ * * * "'1 t -*- I n+r-1, X+r... r, 1 n+r-, al+r-2... a,, 1.............................. ant7+r-1, an+"-... an, 1 | whicl' is of the form of that in Art. 2, may be expressed as the product of three factors. First the product of the differences of all pairs of the quantities i.... xr, i.e. t (x,.... ), which by Art. 2 can bo expressed as a determinant. Secondly, the product of the differences of,all pairs of the quantities a,... a,,, i.e. 3t (a,... a,). And, lastly, the product of all such quantities as f(xi) = (xi - ai) (xi - a,)... (xi - an) = xZi - pxi?1- +... + (- 1)n-k ppn-kXk+.. Hence its value is x- |, x1'r-2... x, 1 ( (ax)..(.) f (Xr).......................... I X r-l, Xr-2... XZr i Multiplying the ith row by f(xi), and then equating coefficients of xlU. x2. x8w..., we get the theorem: If DU,v,w... is the determinant of order n formed by suppressing the columns containing the uth, vth, wth... powers in the array al+, al+r-2... a1 il.*......................... ann+r-1, anli+r —... an, 1, then Du,w,... =! n-u+r-i, pi-u+r-2... pn-u |i (a,, ' " a'),~ Pn-v+r-i, Pn-v+r-2 *- 'Pn-v where pk is the sum of the products k at a time of a,... a,. 9-11] RATIONAL FUNCTIONAL DETERMINANTS 155 If k is negative or greater than n, pk = 0, po = 1. 11. Let us consider the determinant 1 1 1 D=; |Xi - l Xi -2 i -an 1 1 1 i i i X2 - ~l 2i - 12 22 - n' Xn?- - ai X it —2 Xi an Multiplying the ith row by f (xi) = Ui = (xi - ai) (xi - a)... (xi - a,,), we get u1u 2... UnD = — I -'i- k The determinant on the right is an integral and alternating function both of the quantities x... x and of al... a,. Hence by Art. 3 it divides by (x,, x2...,,) ~ (a1, a..... a,). Comparing the orders of the determinant and this product we see they are the same, hence the additional factor is numerical only. To determine it, put x1, 2... x, equal to a, a2...an respectively; then all the elements except those in the leading diagonal vanish, and - = (Xi - 1a) (Xi - a)... (xi - ai-) (Xi - ai+1)... (i - an) Xi - CCi = (- 1)i- (a - ai)... (ai-x - ai) (ai - ai+l)... (ai - an) when xi = ai, thus the determinant reduces to n (n-l) (-1) 2 (, a... a,), which determines the factor. Hence,(n-1) D ( 1) 2 (w1, 2... 2 n) - (al,.2.._ aL) U1 t2... 'Utn 156 THEORY OF DETERMINANTS [CHAP. XII. 12. If Dik is the complement of in the determinant Xi - Ctk D, then Dik is equal to the expression obtained by omitting xi and ak on the right, and multiplying by (- l)ik. n(n-l). Dik - (- )it+k (- 1) 2 (X1 '.. * -lX1 +l... L (al... a k_-lk+l.. al.) 7) V2 ~. V*1 - where i1 U'2 ' Ui-i Uti+l Un vl v2... V1 -l X1 - ak2k x -k XG? _1i- ak Xi+l -a Ck Xi C-ak Now if we write g (Z) =( - (Z - X 2)(Q) - 'x... (z - X,) (x1... xi-lx +i... X,) (ai... ak-1 k+... an) (- )i+k (xl... X) ' (aL... *an) g' (xi)f (k) (x1 - k2 ) (X2 - ak)... (X_ - ak) (xi+ - ak)... (x,( - )= - 1 ) g Xi - kct +hen Di- then D -k - f(:') g- (ak^-) - D- f '(aCk) g' (xi) i- ak 13. The preceding article enables us to solve the system of equations y + +...+ -_ =I x1 - a, X - - n Y_ 3_ YL_+...+.-y_ -t, Xn - Oli Xn- 12 Xn - en viz. Y g (ak) (f(Xi), f (x/ ) U, (lk) f g ( X,) X1 - 'ak 9 (xn) xn - ak} In particular, suppose that u = l =... = z. Then since by the rule for resolving a rational fraction into partial fractions f(x ) - a) (x - ~)... (x- a) g (x) (x - ) (x - x2)... (x- ) = + gf (xi) g' (xi) x - x' 12-14] RATIONAL FUNCTIONAL DETERMINANTS 157 we see by putting x = ak in this, that f (X1) 1 + (Xn) i -... — 1. g (xI) i - ak " g (Xn) Xn - Hence if u-i, Yk (k) 14. If in the determinant D of Art. 11 we expand each term in a series as follows 1 1i ak akp _.lt+-+...+ +-. Xi -k xa Xi Xi P+ we see that the term in the expansion of the determinant which multiplies (xp+l. x2q+1... xnS+l)- is allq, a2q... Ganq xis, %2s... ans To expand the right-hand side of the identity at the end of Art. 11, we have i i i Hl H, - - +4 + — — +... =n n+ +.... Here H, is the sum of all the homogeneous powers and products of order r, which can be formed from the quantities ati, a2... anNow (x1, 2... Xn) = 1-], X-2. x1, 1 X2-l X2-2... X2, 1.......................... Xn-1, n-2... Xn 1 Multiply the ith row of this determinant by the expansion of ui-1; the coefficient of (Xp+l. x2q+... xnS+l)-1 is Hp, H-_.... Hp+,-n Hq, Hq-i... Hq+-n 1q, Hq_8 - q+ln - -1fS; HS_1.. Hs+i-n 158 THEORY OF DETERMINANTS [CHAP. XII. whence we get the final equation P,'l=(__]) 2 H (...,.............................................. cap, aq... (1n0 H~p+_1-, Hq+11-i... Hs+1-n with the convention that H0 = 1, and H, = 0 when r is negative. 15. As an example of Art. 14, a4, a, 1 =- H, H,, H o, c a, a, 1 b4, 6,b 1 H3, H0, O b2, b, C4, c, 1 H,2, 0, 0 c2,, 1 = (a2 + b2 + c2 + bc + ca + ab) (b - c) (c - a) (a - b). We may make use of the results of Arts. 14 and 10 to evaluate determinants whose elements are sines and cosines. For example take X= 1, 1, 1, 1 cosA, cosB, cos C, cos D sin A, sinB, sin C, sin D sin 3A, sin 3B, sin 3C, sin 3D Write for the sines and cosines their exponential values, and suppose eë = a, &c. Then, writing only the first column of the determinant, 1- 1 a1? g=-2i i - i a3 3 a +a- 2 (abcd)8 a4+ a2 a - -1 a4 -a2 a3 - a-3 a6- 1 Add the second row to the third, divide by 2 and subtract the third row from the second, thus i_ __ a3 X = 4 (abcd)3 a2 a4 a6- i 14-16] RATIONAL FUNCTIONAL DETERMINANTS 159 Thus 4(abcd)3 X a2 + 1 a3 a2 a4 a3 a6 a4 where the first determinant =a2b2c2d2 1 = a2bcd (a - b) (a - c) (a - d) (a + b + c + d) a (b - c) (b - d) a2 (c-d) a4 by Art. 8. And the second, in like manner, is equal to (a - b) (a - c) (a - d) (bcd + acd + abd + abc) (b - c) (b - d) (c - d). Hence (a - b) (a - c) (a - d) (b -c) (b- d) (c - d) X = X 4 a3b3c3d3 [a2b2c2d2 (a + b + c + d) + abcd (a-1 + b-' + c-1 + d-l)] i a-b =la. _ /abcd(a+ b + c+d)+ - + ( ++l. V4.a. \/abcd \ b c dl Hence if 2S = A + B C + D X =-25. sI in (A- B) [cos (S + A)+ cos (S + B) + cos (S + C) + cos (S + D)]. 16. If we differentiate the determinant of Art. 11 with respect to xi, the elements of the ith row become -1 -1 -1 (i - l)2 ' (Xi - 2)2 ' (i -a,)2' And thus ( dxdx... dcx, (Xii ak)2 =B. 160 THEORY OF DETERMINANTS [CHAP. XII. We shall now shew that B 1 1 1.D Xi - ti - a2 X -n an X2 -l X2 -a2 X2 --................. o.............. 1 1 1 Xn -a - $x2 $n - an where { } means that the function on the right is to be formed like a determinant, only all the signs are positive instead of alternating. Multiply the ith row of B by u2, then (nlu2... Un)2 B = ( i )2 j(1). The determinant on the right is an integral and alternating function, both of x1, x2... x, and of al, a2... an, hence it divides by t (xi,... X.) *S (a... an). If the quotient is qb (x1, x,... xn), this is symmetrical with regard to each of the variables, and of order n- 1. Thus B n(n-1) 2 (1X, x... *n) D \ lU2... n Now, by repeated use of the rule for resolving a fraction into partial fractions, S (x 0 =?z) (* xi... Xn) K... Xn) f (X) ~f (ci) (xi- i) (ai, X^... -X) (a^ ^^i, COk '*- n) f (w) k f' (Ck) (X2 - ak) and we get finally (Xi>, x... Xn) u1u9...un _f ( f'__ (C) ft_ (aOt, k... ) (2). f' (a)f' (ak)... (r) (1i - ti) (-2 - ak))... 16-17] RATIONAL FUNCTIONAL DETERMINANTS 161 Now, in the first place, in the combination i, k... p, no repetition can occur, for in the product B (ui... un) (X (x1... ) i (aL... an) not only B, but also {/(xx)} vanishes if xz and x, both coincide X2 - X1 with ah. Hence on the right of (2) we must write for i, k... p all permutations of 1, 2... n. Now if we write ai, ak... ac for xi, x2 *... x respectively, only a single term of (u,... un)2 B remains, viz. + If' (ai)f' (ak)... f ) t 2 while (.x, X2... l )= (ai, E2... ) = (a = + C (~a, 2... a), the ambiguous sign being the same for both. Thus (,,... f' (a )f' (Ck)... f' (p)]2 (, ak... ap) 2,2. a) n (n- 1) = (- /) 2 f/ (ai)f' (ak)... f' (ap). Thus B 1 D - (Xi - ai) (x2 - Xk)... (x -- ap) where i, k... p is to be a permutation of 1, 2... n. This proves the theorem as stated at the beginning. 17. The coefficients in the expansion of the rational fraction 1 + bx + b2x+... 1 + aLx + aCx2 +...) in ascending powers of x can be represented as determinants. Viz. if the expansion is 1 + Pl + P2 +... we have (1 + bx + bx+...) (1 + Px + P...)(1 + a + a2x +...), S. D. 11 162 THEORY OF DETERMINANTS [CHAP. XII. and hence equating coefficients Pl = b1 - ai alP + P2 =b2 - a2 a2.P + alP2 + P3 = b3 - a.... o.................................. an_- Pl + a_2 P2 +... P= b - a,,, a system of equations to find P,. The determinant of the system is unity. Hence if, after solving by II. 7, we move the last colunn to the first place and change the sign of this column, Pn= (-1)n a1-b, 1 a2 - b2, ai, 1 a3 - b3, a2, a,, 1...............................1 an - bn, an-_ n-2...... a =( 1)n 1, 1,.,.,. bl, ai, 1,., b2, a2, ai, 1, b3, a3, a2, ai, 1 as we see by subtracting the first column from the second in the latter determinant. CHAPTER XIII. ON JACOBIANS AND HESSIANS. 1. IF yi, 2... yn be n functions of the n independent variables xi, x,... x,, and if = dyi aik dxkyi then the determinant I aik is called the Jacobian of the functions yi... y, with respect to the variables x1... x T. The name was given,by Sylvester after Jacobi, who first studied these functions. The notations d (x,, x.... X r) df \i, y2... 'J(y, y2 yn) have been employed for Jacobians, each of which has its advantages. The first renders evident the remarkable analogy between Jacobians and ordinary differential coefficients. The second is useful when there is no doubt as to the independent variables. If the y's are explicit functions, the Jacobian is formed by direct differentiation. 2. If the functions y,... yn are not independent, but are connected by an equation, (y,) y..* y,) = o, the Jacobian vanishes. For if we differentiate this equation with respect to Xk, we get dp dy, dpd dy2 df dyn dyl dxk dy2 dxk c dyn dxk 11-2 164 THEORY OF DETERMINANTS [CHAP. XIII. where k = 1, 2... n. Eliminating dp df df dyl' dy,'" dyn' from these equations we get (xi. 2) d(yl, Y2... Yn)= d (x1, x,... x) 3. If the functions y are fractions with the same denominator, so that Ui u udyi dui du dxk = dxkc dxk Thus u ld (yl.. Yn)_ u, 0, O d(1i... xn) dul du du, du dx dx- dxcn l dxn....o.o,.................................. du1, du dun du 1l, - - C, n... U - - l dxl d U dxnx dxn du Add the first column multiplied by - to the (i + 1)st column, dxi and we get n+i d (Yi n) U U du... du d (x1... xn) dxu dx~ du, du1 u1, u - dx, dxu, de d whence dividing each of the last n columns by u d (i... yn) 1 du du d (xi... Xn) +l ' dx, '" dx, dut du, dx, '" dxn dun dun U, dxl dxz 2-6] ON JACOBIANS AND HESSIANS 165 4. The determinant on the right has been denoted by K(u, uL... un). It has interesting properties of its own. For example, since the Jacobian vanishes if the quantities yi... yn are related by an equation, it follows that K (u, u... un) = 0 if a homogeneous relation exists between u, u1... un. Vi If Ui = -7, it is readily shewn that K (U, Ui... Un),,- K(v, v... Vn). 5. If the functions y,... yn possess a common factor, so that yi = uiU, d(yi... yn) 1 i, 0, 0 d (xI... n) u du, du du, du U1, U 1 d1., dxn......................................... dun du dun du ',, d n + dx d.x n dxn du In this determinant multiply the first column by d-, and subtract it from the (i + 1)st column, then d (y.. yn)= -l) u -du/,du d (xi... xn) dx dxn du1 du, dx ' dn dun du, I, dxl '" dx, 2 ttn d (i,, u2 *... U~n) _ ( = 2~2d (u1, 2... Uf,) _- KI (u, u1... un). d (x1... Xn) 6. If the functions yl... yn are given only as implicit functions of x1... n by means of the n equations F1(y,... 2Yn, Xi... X) =0,... F ( yn, X1.. X,,n)- 166 TIEORY OF DETERMINANTS [CHAP. XIII. then d (y,...* yn) _1) d (Fi... Fn) d (F... Fn) d (x... x) d(-... x) d(y... yn) For if we differentiate the ith of the given equations with respect to Xk we get dFi dy dFi dy2 dF, dycn dFd dyl dxk dy2 dx ' dyn dxk- dxk' Thus by the rule for multiplying two determinants (v. 4) ( Fv ddF F - F dy or ( 1)nld (Fi... Fn) d (F1...Fn) d (y... yn) d d ( y,..... yn) ' d (i... x) ' which proves the theorem. (i) If Fi does not contain x1... xi-_, then in the determinant d (F1... Fn) d (x... xi,) ail elements below the leading diagonal vanish, and it reduces to dFl dF2 dFn dx~ ' dx2 ' dxn (ii) If F = - +f (x... n then = (- Y d (y,... yn) ~~~aand d(yl*... yn) d(fi... f) d (x... Xn) d (... Xn)' (iii) Suppose that from the given system we deduce by elimination y = 1 (2x1, )2... X*,) Y2 = 2 (Yi)> X2...' n) y3 = 3(~Y1, y2, >3... Xn).o....................... yn =n (Yi.. - n-1, Xn). Since dfi dyL di dyi_ di dyi dyl dxk dyi_ dxk dxk dxk' 6-7] ON JACOBIANS AND HESSIANS 167 we have dal dc d+f = 1, O, O... d(y... y.) dxe ' dx ' dx" d (x... ~x) d^2 d^2 dy' O dx~ dxS dp c h^ 1,... d4 dyl ' dy2 o, ' dx,.........................oooo..o,............. It follows then that d (yI... yn) C_ d1, 2 dp. d d (xl... Xn) dxl ' dx2 '" dx' thus if d (Y y) 0 d(x1...,) we must have da dp2 d = 0, dx1 dx2 "' dxn i.e. we must have d = 0 dxi where i is some number between 1 and n. Hence ij does not contain xi. That is to say, we have Yi -= i (Yi... Yi-1, xi+l... xn). Now yi+i = i+l1 (Yi '. yi, Xi+i.. XZ), and by eliminating xi+ between these we obtain yi+l == 'i+l (Yi... Yi, Xi+2... Xn), so that yi+, does not contain xi+,. Similarly we can shew that yi+2 does not contain xi+,, and so on; finally y, is independent of x, or y, = *n (Yi... yn-i). So that if the Jacobian of y... yn vanishes these functions are not independent. This is the converse of the theorem of Art. 2. 7. If zl... Zn are functions of y... y,, and these again functions of x,... x,; then d (z... z) d ( z,... Zn ) d ( y,... yZn) d (i... xn) d (yi y*)' (x ~..~. a,)' 168 THEORY OF DETERMINANTS [CHAP. XIII. For since dz dzi dy + dzi dy, dzi dyn dxk dy, dxk dy2 dxk dyn dxk we have dzi dzi di yi dxk dyk dxk which proves the theorem. In like manner, if zl... zm are given as functions of y,... yn, and these given as functions of x1... x,r; then d (zi... z) 0 d(- ZZm) = 0, if m > n. d (x1... Xm> But if n2 < n d(z1... z ) d (z, Z,...Z n) d (t, YL Y *..) d (.x... xm) d (yt, yu, y,.v)' C (Xl, x2...,n) ' where for t, u, v... we take all m-ads in n (v. 3). 8. If fi...f, are independent functions of x.. x,, then zi... xn are independent functions of fi... fn, and we have d (fi. fn) d (x1... x?) d (xi... xn) 'd (fi..fn) For differentiating fi with respect to fk we must consider x1... x, to be functions of f...fn. Thus df, dx,1 d d x ddf dfi x dx dfk dJx dk dxn dfk is equal to unity or zero, according as k is or is not equal to i. Hence dfi dxi dXk dfk For in the product only the elements in the leading diagonal do not vanish, and these are all equal to unity. d dfkxi nd Ai Bi are the complements of and in these two deterand Aik, Bik are the complements of - and dwx, in these two deterdxk dfk minants, we have dxi dfi A = Aki, Bdf =, Bk dfk dXk 7-10] ON JACOBIANS AND HESSIANS 169 Also A (X.. = m) d (fm+...fn) d{fl'*fm) rm+i...) Z d(fi... fm) d (xm+... Wn) Bc (.fI..f)A (X~+... x) d (x1... xn) d (fM+...fn)' For we have just seen that dfi dx, df dx, df, dxn dxî dfk d xc dfk dxn dfk............................................. dfk dx, df ddx dfk dx,n + 2-+ + =1 dx dfk dx dfk df dxn djf............................................. dfn d + dfn dx+ dfn dX -+-f + + = 0. dx, dfk dx dxfk " dx, dfk Multiply these equations by Ai, Ai... Ani respectively and add, then (iv. 11) dxi A = Aki. dfk Similarly we can shew that B dfi =-Bki dxk Again we have (vi. 5) An... Ai,.All A. d (fm+l.fn)............... = Am-1 d n d (X+... x) AMI... A, Substitute in the left for Aik the value just found; thus d (x... m) d (/+... fn) d (fi... fr) d (fm+. * n) which on dividing by Am-1 gives the result required. The last equation is proved in a sinilar way. 10. If we suppose the functions fi...fn to depend on t, we have (iv. 16) dA = Aik dtdf ( =1, 2... n), dt 11tIXk 170 TIHEORY OF DETERMINANTS [CHAP. XIII. and Aik = A dXk dfi' dA _ A f, d d2fi d 1i dx2+ 1 *dt \dtdx, dfi dtdx, dfi dS (dfi) d d Id/N dt. log A = dfi \dt A similar relation holds for B. 11. The relations between Jacobians present great resemblance to the ordinary formulae in the differential calculus. Thus the formule d (z,... z,) d (zi... z,) d (yl... yn) d (xi... n,) d (y,... y'n) d (xi... Ol)' d(fi n.A) (x(i... x)= 1 d(xi... x)' d(fi...f,) are the analogues of dz dz dy nd dy x dx dy dx d dxdy This analogy, which was perceived by Jacobi, led Bertrand to devise a new definition of a Jacobian. Let fi...f be n functions of the variables x,... Xn. Now if we give to the variables n distinct series of increments d1xi, dx2... d2xn d&x1, d>2x2... d2n (1),...................... dn xl, dn x2... dnixn let the corresponding increments of the functions be d1fi, dif2... difn d2fi, d2f2... d2fn (2)........................ djifi, dnfg... dnfn Thenjust as the differential coefficient of a single function of a single 'variable is defined to be the limiting ratio of corresponding incre 10-12] ON JACOBIANS AND HESSIANS 171 ments of the function and variable; the Jacobian of the functions fi... fn of the n variables x1... x, is defined to be the limiting ratio of the determinants of the systems of increments (2) and (1). That this leads to the sane Jacobian as before is plain from the equation dkfi = dfi- + dk +.. + dfi kn dx c4x, dx, which gives (v. 4) \dkfi-= dkX. df., odk.fi _ d (fi... fn) \dkXi d(X... Xn)' according to our former definition. Using this new definition we can prove all our former theorems. Let us use it to prove the first of the above equations, viz. the theorern of Art. 7. If the system of increments given to x1... x* be C1x1... dlxi.oo........ dn xl... d xCn n let the corresponding systems for y,... yn and z1... zn be dyi... dyn dlz... dlzn.oooo..... -........................ dy y... dnyn dn z1. dnzn Then we have identically dizk_ Idizk\ Idiykl diXk - \diyk' \dixk l or by definition, d (z... n) d (z... Zn) d(yi... yn) d (x... Xn) d (yi... yn)x' d (x * Xn) 12. We can also, using alternate numbers, obtain a symbolic expression for the Jacobian, from which the ordinary results follow. Viz., yi... y,, being n functions of x... xn, let y = ely + ey +... + enyn, X = eix, + e2x2 +... + ex,. 172 THEORY OF DETERMINANTS [CHAP. XIII. Then dy _ dyl dy2 dyn e a + e -- + '.. + e. dx dxi dxi dxi' whence (II. 15) dy dy dy_ dyl dyl dx ' dx''' dx, dx, dx' dyz dyn dx, '". dx? d (y,.. ) y) d (i... x,) But now dy dy dx dxi dx dxi dy = e~i d Thus the above equation (1) becomes /dyY _ d (y.. ^yn) cdx; d (xi... xn)' from which symbolical equation we can deduce our former theorems. For example the equation (dy} (d )l =1 \ dx) \dy) gives at once d (/i... yn) d, (x... n) = _ d(X1... Xn)' d (y... yn) 13. Jacobians occur in changing the variables in a multiple definite integral. Let us transform the integral I ==... F (y... yn) dyl... dyn to an integral with respect to x,... x,, the functions yi... yn being supposed given functions of x,... xn. We proceed in the manner used by Lagrange to transform a triple integral. Beginning with yn we have to find the sum of the quantities Fdyn, 12-13] ON JACOBIANS AND HESSIANS 173 while yi, Y2... yn- remain constant. This gives us ~ dx, + d 2 +..:+ dl dn dx, dx, dx2 dyl dy _ dyd o= dY dx, + i2 dx, +... + d.. dx, dxy dx, dxy... o,, oo,,,,oo........o........... dy, cy dy dyn = dx d + dx +... + dxn. dx, dx, d~n Solving this to find dxn we get (xi. 1) Jn-, dyn = J, dx, where Jr d (yl, y2... y,) Hence we must replace dy, by - dx,, and n-i I=...Fd... dyn.. F J dyl... dyndxn, the limits of xn being determined from those of yn. In this integral begin by integrating with respect to yn-. We have to find the sum of the quantities F j- dyn, while fn-i y... yn-2, xn remain constant, so that dO Y dx, + +... d dx d... -i dy =/y2 dx +... + d dx, dx._-.................................... dyn- l= d1-dx, +... + dXn-l, which gives Jn_ dyn-i = Jn, dxn-_ Thus dyn_ is to be replaced by Jn_- dx,_,, and F J- dynn-2 n-1 174 THEORY OF DETERMINANTS [CHAP. XIII. J Jin_, by F. j. -l dx,. Hence, the limits being properly detertn-i n î^-2 mined, I... F J-n dy1... dyn,_dx.ndxn. fn-2 Similarly if we began by integrating this with respect to yn-_ we should get a system of equations which would give us dyn-2 - dx=n-, Jn-3 and I =... F dy... Fdyn-_3dxndxndxn. J Jn-3' Proceeding in this way we should finally obtain I =... F -l dyldx,... dx,. Then we integrate with respect to yi, subject to the equations dx, = O, dx = 0,... dxn = O, dyi so that we must replace dyl by -Y dxl, i.e. Jldx1. Thus I =... FJndxldxi... dxn _ I. * F (x) t (y., y2.. yn) d d x *. dxn, -j.. i ^ d (,.... yn) F (x) being the result of substituting in F for y... yn their values in terms of x1... x. 14. As an example let us consider the following determinant of definite integrals due to Tissot; we shall however follow Enneper's proof. Let a,, a... an be n constant quantities in ascending order of magnitude, and let ~m (xm) = (Xm - a)'1 (m - a)P 2... (Xm - a,)rm (am+i - Xm)Pî)n+... (a. - xm) where pl, p2... pn are either positive proper fractions or any real 13-14] ON JACOBIANS AND HESSIANS 175 negative numbers. The determinant to be considered is then ) = J *... Jin T1....... Jni *.. Jnn where +l Xk-1 e-dxk Jik = * dXk (a,+ = oo ). Thus -D=(- ) 2 I dx dx... dx (x...... J 2 an s1(X1) a2 (X2) J. * 2 (an) (exp. U = e). Now let us introduce in place of x1, x... xn the n new variables y... yn, given by the equations Yi + _..y Y =1 x1- a1 - a2 1 - an Yi + Yn _ -1-... - - Y- = 1. -,a- a - x n- an Then by xII. 13, g= (ak) Y~k = ' (ak)' and hence dyk yk dxci xi- ak' Thus by xII. 11, d (y... yn) 1 d (x, x... xi - ak Y( - 1) yl. yn (xi..(. X) (a1... an) f (x) f (x)...f (x) ~o w y~... y g (ai)... g (a) n ('+ Now Yi Y-g (a (-1) 2 d (y... y,) _ ~ ((... an ) therefore yn)...n) d (xi... Xn) 2(al... an) 176 THEORY OF DETERMINANTS [CHAP. XIII. Hence in the integral we replace dx,... dx,, '(x... x,) by dyl... dyn g' (al... an). Now if we write Fm (Z) = (z - a)... (z - a) (a+,,- z)... (a, - z) we have Y Pl () 4 (x3)... b11 (X) y1Piy2l * Fi' (a,)P' F,' (a2)P... T1' (a.n)' Hence... (zx... x1 ) dx d..x 01 (Xl) ~~.(n) 1 ~ (x3)... (x) is replaced by dy... dyn C (aL... an) ylPi... Y.Pn T1' (a1)Pi... (. Fn' (an)Pn Again xz... x, can be regarded as the roots of the equation Yi y 2 yn + + +... - 1, Z- a1 - 2 Z - a,, the roots of which lie between a, and a2; a. and c3;... a, and oo. Hence y,... yn take all positive real values. Also we have xi + 2 ++... + xn = y +... + yn + ai +... +a. Thus our integral reduces to n (n-1) (-1) 2; t(aC1... a,,) exp. (- a-...-a,) F-' (a)pl... Fn' (a,,)Pn j... exp. (- y-...- Yn) dy1. dyn f:'" y -... YiPlL z(n-l) (1) 2 r (1-pl) r (1-p -)... (l- p) e —_-...-^ { ' ((ai)2-~... ( ) -} 15. If u be a function of n variables x,, x2... x,, and yi... yn its differential coefficients with respect to these variables, since dyi _ d d( d2u dxk dxk \dxi dXdxi- ik 14-16] ON JACOBIANS AND HESSIANS 177 the Jacobian of y... yn is a symmetrical determinant formed from the second differential coefficients of u. This determinant is called the Hessian of u (after Hesse), and is denoted by H (u), so that H (u)= Uik\ The Hessian of u will vanish if the first differential coefficients of u are not independent (Art. 2). For example, if U = X12x22 + X12 X32 +... + Xi2k2 +... + X2n-l' 2, d2?.u d = 2 (x2 +...+ xi_ + 2i+l +...+ 2), d2u = 4XiXk; dxidxk.. (u) = 2 (X22 + 32 +.. + n2), 4x1x2. 4x1x2, 2 (xi2 + 32 +... + Xn2)... '.......................................................... Or, dividing the ith row by 2xi and the cth column by 2xk, H (u) = (2^x1ox2... x*)2 x22 + x +... + x2 2xi2 n,..i 2 + 32 +... + n2 222 This is a determinant of the forin of that in IV. 25. If we write 3o =2+ x2 + 2 +... + Xn2 V=( - 12) (O- X2)...- (-n2) H (u))= 6nV + E _ X.2} If u = X2y2 + y2z2 + z2x2, this gives I (u) = 24 {9x2y2z2 - (X2 2 y2 + Z2) u}. 16. Jacobians and Hessians belong to the class of functions known as covariants. That is to say, if these functions are S. D. 12 178 THEORY OF DETERMINANTS [CHAP. XIII. transformed by means of a linear substitution, the Jacobian of the transformed functions is equal to the Jacobian of the original functions multiplied by the modulus of the substitution, and the Hessian of the transformed function equal to that of the original function multiplied by the square of the modulus. Let the variables be transformed by the substitution xi = ailel + ai22 +... + ain (i = 1, 2... n) and let the functions y,... yn of x1... x become in consequence the functions y/', y... yn of et...:. Since dyi' dyi dx, dyi dx2 dyi dxn dk dx~ dx k dx2 dck dxz d k dyi dyi - d alk --... + d ank, dx1 dx-n it follows from the multiplication theorem that d (yl /.'. yn) d (y,... yn) d (1... n) d (x... xn) which proves the theorem for Jacobians. To prove the theorem for Hessians, let u be the original and u' the transformed function. Then since the Hessian of u is the du du Jacobian of -u d we have dxl dxen (du' du' du' ~ dÇ1,.2 \^ W) H (U') d e7l d2.. d ) d(), 2... n) d du' du' d (x...X) Xn d2u' d2 Now Ndxidtk d:kdx,, / du du.H (,d') dx. dx I de1:)...-) l) d, d dt a d_ "(u'- dxj / = d(x,...^) aik} =- H((). laik|2. 16-17] ON JACOBIANS AND HESSIANS 179 17. If we have n linear functions yi = biùx, +... + bi (i = 1, 2... n), clearly d (,.. bik d (x,... xb) Il If u is a quadric function U = blxi2 +... + 2bikXiXk +..., then H (u) = 2n bik, (bik = bki) The symmetrical determinant on the right, which is called the discriminant of the quadric, is therefore an invariant which on transformation is multiplied by the square of the modulus. 12-2 CHAPTER XIV. APPLICATIONS TO BILINEAR AND QUADRATIC FORMS. 1. A BILINEAR form is an expression which is linear and homogeneous in each of two sets of independent variables. If the number of variables in each set is n, any such form is defined by an equation A = aikxiyk. (i, k =1, 2,... n) If B is another bilinear form with coefficients bik, a third form C can be derived from A and B, with coefficients cik which are the elements of the matrix (an) (bnn). Thus C= Zailblkxiyk (= 1, 2,... n) aA aB = _ It is convenient to write symbolically C=AB; it will be observed that AB is, in general, different from BA, so that the multiplication is not commutative. But it is associative and distributive; thus, for instance, if P, Q, R denote any three forins, P(Q+R)=PQ+PR, P.QR=PQ.R. In the particular case when AB = BA, the forms A, B are said to be commutable. We have a series of forms represented symbolically by positive integral powers of A; these are commutable, and obey the ordinary laws of indices. 1-3] APPLICATIONS TO B1LINEAR AND QUADRATIC FORMS 181 2. With any form A are associated the matrix and the determinant of which the coefficients of the form are elements; and we have (AB) = (A) (B) with a corresponding theorem for the determinants. The form PAQ may be derived from A by a linear transformation of each set of variables. For if we put a P, aQ i = ^ I= -p7likl, Xk= gQ = qkmym the form A (~, 7/) becomes ap aQ ap a_ m aQ, yi aXk ay axi ayk aXkJ P. AQ = PAQ. If the substitution is cogredient, pl = qil, and the matrices of P, Q are conjugate (v. 1): in this case we shall write Q =P' and call P' the conjugate of P. 3. The form E = iyi (= 1,2,...n) is called the unit form. If A is an ordinary form, that is to say if A \ does not vanish, there is a form A-1 such that AA-1 = A-A = E. This is proved by assuming A-l= pikXiYk and equating coefficients. Clearly A pik = Oik where aik is the coefficient of aik in A. The form A-l is called the reciprocal of A; its reciprocal is A itself, and if we adopt the convention that A~ = E the laws of indices hold for all integral powers of A. Again (ABC)-i = C-1 B-lA-1 and similarly for any number of factors, if all the forms involved are ordinary. 182 THEORY OF DETERMINANTS [CHAP. XIV. If AB vanishes identically, either A = O or lB = 0; in particular, if A is ordinary, B = 0. because in this case B=A-.AB =0; hence the ordinary theory of equations may be applied to symbolical polynomials involving powers and products of ordinary commutable forms. For instance, A2-E=A2-E2 =(A +E) (A - E), and if either of the factors on the right hand is ordinary, the other must vanish identically, if A2 = E. 4. A form A in which aik=0 except when i=k may be called a normal form. Supposing that the coefficients belong to a field with the properties stated in vIr. 2, it follows from vII. 10-12 and Art. 2 of the present chapter that rational unitary forms P, Q can be found such that PAQ =N where N is a normal form. The number of terms in N is equal to the rank of | A I; we shall call this the rank of A. The determinant of A and its elementary factors are invariants of A. If A, B are any two forms, and X an indeterminate, the result of equating to zero the determinant of XB -A is an equation in X, the roots of which are invariant for simultaneous transformations of A and B. The most important case is when B is the unit form; putting XE-A = b (X), b (X) is called the characteristic function of A, and b (X) = o the characteristic equation of A. 5. Any rational function of a variable t can be reduced to the shape g (t) h (t where g (t), h (t) are polynomnials. If, now, A is any bilinear form, 3-6] APPLICATIONS TO BILINEAR AND QUADRATIC FORMS 183 g (A) and h (A) are the symbolical expressions of two forms derived from it. If h (A) is ordinary, the form reciprocal to it will exist, and we may write / (A)=g(A) [h ()]- = g (A)/h (A). Let g (t) be a polynomial in t, with roots ti, t,... t,; and let (X)= (X) - X,) (X - \X)... (X - X,), b (X) being the characteristic function of A. Then if g(t) = c (t- t)... (t - tm), g(A)= c (A - tE)... (A - tE), and hence Ig(A)l=c"lA -tEl... A-tE I = (- 1)"cq( (tl) q (t2)... (tm) = (- 1)mCn n (th- Xk) = (X,) g (x2)... g (Xn). Similarly, if f(A) is a rational function of A of which the denominator is ordinary, If(A)! =f(X)f(X)... f (X,). Changing f (A) into XE - f (A), which is also a rational function of A, XE -f(A)| = I{\ -f(Xi); hence the roots of the characteristic function of f(A) are f/(X), f(X2),... f(\n) where Xi, X,,... Xn are the roots of the characteristic equation of A. As a particular case, the form b (A) has a characteristic function Xn, and the roots of its characteristic equation are all zero. As we shall presently see, the reason of this is that J (A) vanishes identically. 6. The coefficients of A2, A3, etc. are rational integral functions of the n2 coefficients aik; hence it must be possible to find Co, cl,... cp, rational integral functions of the coefficients of A, such that ( (A) = coA~ + ci A1 + c2A2 +... + cpAP = 0 identically, for some value of p which does not exceed n2. We may suppose that f(A)= O is the equation of lowest degree which is satisfied by A: thus c, does not vanish. 184 THEORY OF DETERMINANTS [CHAP. XIV. Consider the equation Ao A A2 S + - + +... r r2 r' where the symbolical expression on the right is an infinite series, and r is an ordinary numerical quantity. By taking r large enough, the form S is interpretable, because its coefficients are convergent series in r. Multiplying by J (r), we obtain, (r) S = G (r) where G (r) is an integral function of r with coefficients which are integral functions of A; all the negative powers of r disappearing, in virtue of jr (A) = O. Thus S can be represented as an integral function of A with coefficients which are rational functions of r. Again, A' A' rES=AO~+- + +-... r r2 =A~+ AS; consequently (rE -A) S= A = E and S=(rE-A)-'= F(r) ~ (r)' where ) is the characteristic function of A, and O 1, ) X2.. Xn F(r) = y1,, a- r, ai,... al,, = afikxiyk,................................ y, an,, an,,.. ann - r fik being a first minor of A-rE. Since F(r) G(r) and * (A) is the lowest function of A which vanishes, 4 (r) must be prime to G (r) for arbitrary values of the variables xi, yi and, in addition, must be a factor of' b (r). Suppose b (r) = (r) % (r); then < (A)= 4(A) X (A) = 0, 6-7] APPLICATIONS TO BILINEAR AND QUADRATIC FORMS 185 and consequently A satisfies the equation ( (A)=O where < (X) is the characteristic function of A. Evidently %(r) is the greatest common measure of <(r) and the quantities fik; in other words (viI. 3) it is the determinant factor D,_1 of < (r) written in its determinant form. Hence also (r) = S (r)/X (r) = the nth elementary factor of <b (r). 7. If A is an ordinary form, the constant term in r(A) is different from zero; for otherwise, we could multiply the equation 4r(A) = O by A-, and obtain an equation of lower degree satisfied by A. Suppose, now, x being an indeterminate, that * (x) = q (x - a)a (x - b)P (x - c)Y..., a, b, c... being all different from zero. We shall prove that there is an integral function of x, say x (x), such that { (x)}2 - is divisible by * (x). Taking </a with a determinate sign, we may write x = {/a + (x - a)} -,/al 2a 8a } = F(x)+ (x- a)- R(x), where F (x) is a polynomial of degree (a - 1), and R (x) is a series proceeding by powers of (x - a), which is finite when x - a = 0. Similarly x= G (x) + (x - b) S (x) = H (x) + (x - c) T (x).. and so on. Now let x (x) - F(x( x -() + ( (x- )y +"; this is an integral function of x, and X(x)- xR ( Rx)+ +x (x - a)" (x - b)P 186 THEORY OF DETERMINANTS [CHAP. XIV. where the right-hand member is finite when x = a. Similarly x ()- \/ x (x)- x/ (x-b)~ ' (x- c) ' * are finite for x = b, c,... respectively. Hence {x (x)2- x x (x)+ V, x (x) - 4\ r (x) q '( - ()" (x - b)l... is finite for all finite values of x, and is therefore an integral function. Since r (A) = 0, it follows that {x (A)}2=A and we may write (A)= U= A. The form U is ordinary, and we may also write U-1 = A-~. 8. Let A, B be any two ordinary forms, and let P= B (AB)-i = 4pikXiyk; then PAP = B (AB)-t AB (AB)- = B, p-l BP- = A. We have therefore found a substitution Xi == "pki k, Yi = pikqk k k which converts A (x, y) into B (~, q). The coefficients of the substitution are rational in the square roots of the roots of the characteristic function of AB. In this field of rationality, then, any two ordinary forms are equivalent. Let us now inquire whether A can be transformed into B by a cogredient substitution; that is, whether a form Q can be found such that Q'AQ = B, Q', as usual, being the conjugate of Q. We shall begin by supposing that A, B are both symmetrical, or else both skewsymmetrical. 7-8] APPLICATIONS TO BILINEAR AND QUADRATIC FORMS 187 Suppose P determined, as above, so that PAP = B; then since A' = + A, B' = + B, corresponding signs being taken, it follows that P'AP'= B; and hence that (P-1P') A (P'-1) = A. If now we put P-1P'= U U' = P (P')-', and UA = AU'. Hence, also, U2A = UA U' = A U'2, and, generally, if X (U) is a polynomial in U ( U). A=Ax (U'). Let X (U)= U-, and let =(U') P'; then Q'AQ = P U'A ( U')P' = PA U'P'= PAP = B; so that two symmetrical, or two skew-symmetrical forms, if both ordinary, can always be changed one into the other by a cogredient transformation, the field of rationality being suitably extended. More generally, if we start with PAQ= B and write (Q')-P = U, then x( U) A=Ax (U'); and if X (U) is ordinary PX (U)-1Ax (U') Q = B. To make this a cogredient transformation, we must have PX (U)- = Q'% (U), or x(U)2= (Q')-lp =. 188 THEORY OF DETERMINANTS [CHAP. XIV. Conversely, if R = (U') Q = (P'Q-1) Q it follows that R'AR = B. 9. Now let A, B be any two forms, A', B' their conjugates, and suppose that, u and v being indeterminates, two forms, P, Q, independent of u and v, exist, such that P (A + vA') Q = uB + vB'. Write A +A'=A,, A-A'=A2, B+B'=B1, B-B'=B2; -B + BI=B,, B - B = B.; then A1, B1 are both symmetrical, while As, B2 are both skewsymmetrical; and moreover PAQ = B, PA2Q=B-. Hence, by the method of last article, a form R can be found such that R'A1R = B1, R'A2R = B2, and hence also R'AR = ~R' (A1 + A) R = B, R'A'R = IR' (A - A2) R B'. Therefore the necessary and sufficient condition that A may be transformable into B by a cogredient substitution is that forms P, Q can be found such that PAQ=B, PA'Q= B'. This result is due to Kronecker: the proof here given is that of Frobenius. The equivalence of bilinear forms, whether ordinary or singular, has been completely discussed by Kronecker and Weierstrass; the subject is too extensive to be pursued here. It should be observed that all the theorems of this and the preceding articles of this chapter admit of a three-fold interpretation, according as we refer them to bilinear forms, determinants, or matrices. The product of two matrices has already been defined: we may call the matrix associated with the forrn A + B 8-11] APPLICATIONS TO BILINEAR AND QUADRATIC FORMS 189 the sum of the matrices associated with A and B, and then the symbolic calculus of forms becomes a calculus of matrices. From this point of view, the subject was initiated by Cayley. 10. We have already considered (vii. 10) the reduction of a matrix to a normal form; the process there given also supplies a method for the reduction of a bilinear form. Suppose, in particular, that the form A with which we start is symmetrical. Then, in carrying out the process of reduction, we may arrange it so that the elementary transformations are made in successions of conjugate pairs. After each such pair the transformed matrix is again symmetrical, so that finally we get a cogredient transformation PAP'= N= eeixiyi where r is the rank of the matrix of A. Let us now make the two sets of variables coincide; then A becomes a quadratic form, of which i a,, ( is the discriminant, and we have the theorem that by a linear transformation of the variables a quadratic form can be reduced to the shape e1x12 + e2x22 +... + erXr2 where r is the rank of the discriminant. The discriminant is an invariant for any linear transformation, because PAP'[I=PI 2IA. 11. The reduction of a quadratic form to a sum of squares may be effected by a transformation which is rational in the coefficients of the form and in a certain number of indeterminates. The possibility of this arises from the fact that the general linear transformation involves n2 independent coefficients, while the conditions that the new form may be a sum of squares are n (n-1) in number, and this is less than n2. Let us write a = aki, i = - = aikXk; (i, k = 1 2,... n) dXi ke and let us suppose that (dnn) 190 THEORY OF DETERMINANTS [CHAP. XIV. is a matrix, with arbitrary elements yik. Consider the symmetrical determinant all, a2... ain Yl yl2... yln Ui; am an2... ann Ymi Yn2 Ynn Un Un= / y 1 yi... yn... O O O yin y2n...ynn 0 0... 0 0 U.... Un 0 0... 0 this vanishes identically, because every minor obtained from the last (n + 1) rows is zero. The value of the minor obtained by omitting the last row and column is (- 1)n ynn 12; we denote this by Rn, and suppose that it does not vanish. By omitting from Un the rows and columns which contain elements yik with i or k greater than p, we obtain a symmetrical minor which we shall call Up. The determinant obtained by omitting the last row and column of Up we shall call Rp. Finally let Xp be the determinant derived from Up by omitting the last column, and the last row but one. Thus Xp is a linear function of uZ, u2,... Un, and therefore of x1, x2,... xn. We shall suppose, in the first instance, that!cnnl is different from zero. By the properties of first minors (vi. 5) Rp Upl —Xp2= - pRp_l, and hence Up-1 UP Xp2 Rp- R e-p R vp-i (1p. Now Un = 0, and Uo = - a,,n u, as we see by multiplying the first, second,... nth columns by x,, x2,... xn and subtracting from the last column. Summing the equations of which (1) is a type, from p = O to p = n, and writing R0 = I a, j for uniformity, we obtain X 2 X 2 X 2 n =- -RRR, - R, _R............(2) which gives the required expression of as a sum of squares. which gives the required expression of u as a sum of squares. 11-13] APPLICATIONS TO BILINEAR AND QUADRATIC FORMS 191 12. At first sight it seems that a transformation has been obtained involving n2 independent parameters; whereas we have seen that this is impossible, the number of independent constants, in general, being -n (n + 1). The explanation is that the constants Yik enter into the quantities Xi, Ri in particular combinations, namely certain minors of | ynn; these are connected by identical relations, and the number of really independent parameters is reduced accordingly. Thus, to take the simplest illustration, let n=aax + 2hxy + by, (y2,)= (v ); then the direct application of the above process gives X = (Xp - iv) {(va - Xh) x + (vh - Xb) y}, X = - (ab - h2) (Xx + vy), Ro =ab-h2, R =-bX2 + 2hX\ - aL2, 2 = (Xp -,)2; and formula (2) of last article reduces to 2 (ab - h2) (XX + vyy)2 + {(av - h) x + (hv - bX) y}2 ax2 + 2hxy + by = --- - 2 bX2 - 2hXv + av2 Here there is only one independent parameter, namely X/v; but instead of X,, X2 we may take arbitrary multiples of them, and this gives two more parameters. The method which has been explained is due to Darboux; he has shewn that it does, in fact, give the most general substitution of the kind required. 13. Let us now suppose that LannI is of rank r; that is to say, let one at least of its minors of order r be different from zero, while all those of higher order vanish. Then, in the notation of Art. 11, Ro) Rl,... R_-n._vanish identically, but Rn-. does not, so long as the quantities Yi remain arbitrary. We shall still have Ri+, Ui - Xi2 = Ui+ Ri for i = n -r, n- r+ 1,... n; and we conclude as before that Un-r _ X2n-+7'+1 X 2 Rn-r -Rn-r Rn-r+i n-i Rnl 192 THEORY OF DETERMINANTS [CHAP. XIV. In the determinant U,_n subtract from the last row the sum of the products of the first n rows by xi, x2,... xn respectively; the last row now becomes 0, 0,... 0,-,-2, l.- In-r, - u, where i =- ysixs. s From the last column of this new determinant subtract the sum of the products of the first n columns by x1, 2,... Xn respectively; the new determinant is symmetrical, and its value is Un-r =- 6Rn-r + -lilkpik, where pik is a minor of Rn- obtained by cancelling the row and column containing yik. But pik =0, because it can be expressed as a linear function of minors of lan, which are of order higher than r: hence Un-r = - Un-r and X 2n-r+l X2n-r+2 X 2 U = - Rn-rRn-r+i R —r+i-ln-r+2 Rn-,lRn Thus in the general case, where r, the rank of lann,, is unrestricted, Darboux's method gives the reduction of u into the sum of r squares. The values given to the parameters Yik rmust be such that none of the quantities Rn-r, Rr+l,... R. vanishes; this is always possible, since they are functions of the quantities yik which are not identically zero. 14. Returning now to the case when l ann does not vanish, we will shew that if 1,,2) - en are aiiy assigned independent linear functions of x1, 2x,... xf, we can, by a suitable choice of the quantities Yik, express the quadratic in the form U = + (aiel + a... +. + an) ~ (b22 +... + bn-, n-l_ + bn )2 +.. ln2 1 w e 2 a la f o, 2 where Es is a linear function of 8s, s+1,... '. 13-15] APPLICATIONS TO BILINEAR AND QUADRATIC FORMS 193 Suppose =ck = Cj Xj, (j = 1, 2,... n) then Icn does not vanish, and the matrix (/3) can be determined by making Ui = ekk (i =1, 2,... n) k identically. In fact, this gives (an) = (Pnn) (cnn), and therefore (/3n) = (ann) (Cnn)-l. In the expressions we have denoted by Rp, Up put yik Aik; and to fix the ideas suppose that Xp bas been obtained from Up by omitting the last row and the last column but one. Let Zp be the determinant obtained from Xp by putting yik = 3ik; then if the columns headed by,11, 812,... /, p- are multiplied by e^, 2,... p-, respectively, and the sum of the products subtracted from the last column, the new elements of the last column, read from the top, are v, v2,... v,, where k=n vi = Y /ikek. k=p Hence Zp, when expressed as a linear function of ei,,... n, does not involve el, 2>...,p_,; and this proves the proposition. 15. The advantage of this transformation is that if we suppose tp+l = ep+2 =... en = 0) so that Zp+1 = Z+2 = Z=, the resulting values of Zl, Z2,... Zp are linearly independent. Hence we get the reduced expression of u as a sum of p squares when the variables are subject to (n-p) linear relations. This reduction is important in problems of relative maxima and minima. There are two special cases which deserve attention. The first is when k = lk, so that /ik = ik, where 8ik, as usual, is Kronecker's symbol. The value of Rp is now (-1l)PAp, where Ap is the minor obtained from a,,! by omitting the first p rows and columns: thus, writing A for ann I, we have (Art. 13) Z'2 Z2 2 = — $+ 2+' + - where Zs is a linear function of us, us+,... un. S. D. 13 194 THEORY OF DETERMINANTS [CHAP. XIV. The second case is when ek = xk, so that 3k = aik. The value of Rp is now (- )1^AA'p, where A'p is obtained from A by omitting the (p+l)th,... nth rows and columns: thus, with A' =A for symmetry, zA 2 z222 2 Az'= + +...+ +. where Zs is now a linear function of x, xs+,.. xn. Sylvester has proved that when a quadric is linearly transformed to a sum of squares by a real substitution, the number of positive and negative squares is always the same. The results of the present article shew that the variations of sign are determined by either of the sets AA1, AA,,... n-2An-1, Aln-, A A,..'2 A' -' An- A'nIn particular, the necessary and sufficient conditions that all the squares may be positive are that either of the series A,, A2,... A, A', AA should consist of terms which are all positive. In the case when the variables are s bject to (n- p) linear relations, the variations of sign in the reduced form of u are obtained from (aq,) (iPnp) (a,)' (0O) and its leading minors, the elements /ik being determined as in Art. 14. 16. If a quadric, by means of a linear transformation, has been reduced to the sum of n squares, u = aik XiXk = A1yl + A2 y22 + + An2yf; the discriminant of the right-hand side is AA... A,,, and hence if /, is the modulus of transformation, A,A,... A,, = fi 1 a,,n Two given quadrics u = CaikXiXk, v = bbikXiXk 15-17] APPLICATIONS TO BILINEAR AND QUADRATIC FORMS 195 can, in general, by a simultaneous linear transformation Xi = CilYi + Ci2y2 +... + inYn (i = 1, 2... n) be reduced, each to the sum of n squares of the saine linear functions, viz. u = A1yf2 + A2y22+... + Ayn2, v = sAys12 + s2A2y2 +... + snAny,2; for in order to determine the n2 constants, cik, we have first n (n -1) equations from the fact that the coefficients of the products yiyk must vanish, and n additional equations from the condition that the ratio of the coefficients of yi2 is to be si; in all, n2 equations. If we form the discriminant of su - v, its value for the original quadrics is 1 sa,- b,n..........................(1), and for the transformed quadrics A1... An (s - s,) (s - s2)... (s -........... (2). The ratio of the quantities (1) and (2) is p/; hence si... s, are the roots of the equation (s) = sa - b = 0..................(3). 17. The following resolution is due to Darboux. If we write diF F=su-v, Xi = ld = -vi...............(4), we have identically by Art. 11 1 sa,-bln... san,, - b,,, X F= - s8 -- V = (s)................................. sa,m- bl... san-bn,,, Xn...(5). X1... Xn The determinant on the right is a function of s of order n - 1; resolve the fraction into partial fractions, and we get 1 sianl - bi.. sia,, - b,,, X1 8., — - '-' --- (Si) ( - Si)................................. si Can,- b,... sian~- bn,,, Xn...(6). ~... Xn 13-2 196 THEORY OF DETERMINANTS [CHAP. XIV. The determinants on the right are all perfect squares by VI. 6, for they are obtained by bordering the vanishing determinant A (si). Whence UT2 su - v = - A (si) (s- si) where Ui is a linear function of the form Ui = d, XI +... + dinXn. If in the determinant (6) we replace Xi by its value from (4), and subtract from the last column the first n multiplied by x1... x, and do the same for the rows, the value of the determinant is unaltered, but Xi is replaced by 2 (s - si) du 2 dx< A term is also introduced in the principal diagonal in the last place, but since its minor vanishes by (3) we may replace it by zero. Thus Ui is replaced by d d du) = (s -i) Vi, where Vi is independent of s; Vi(s -Si).s. su- V = 2 V7.A'(si) Equating coefficients of s we get V2 _ SiVi2 Aà (si) A A' (si)s which is the required resolution. It is assumed here that s,, s2,... s, are all different: when this is not the case, the analysis requires modification. For a complete discussion, the memoirs of Weierstrass, Kronecker, and Darboux should be consulted. 18. An important branch of the theory of quadrics is that of their linear automorphic transformation. That is to say, as the name implies, the discussion of those linear transformations which do not alter the outward appearance of the quadric. So that if x *... x, are the original, and y,... yn the new variables, Saikxixk becomes Saikyiyk. 17-19] APPLICATIONS TO BILINEAR AND QUADRATIC FORMS 197 Without entering into a discussion of the general case we shall study that particular one which gave rise to the whole theory. In the transformation from one set of rectangular axes in space to another with the same origin, the distance of a point from the origin is the same, and expressing this for the two systems x2 + y2 + z2 = x'2 + y'2 + z2; such a transformation is linear and automorphic, and is known as an orthogonal transformation. 19. The general case of an orthogonal transformation is to determine those linear transformations which give us,2 +X22+... +x2 =y +y22+...yn2. The theory is due to Cayley, but we shall give it here as modified by Veltmann. Let us consider the following equations bilxl + b,2x2 +... + bbixn = b11 y + b2 y2 +... + bniyl) b21x1 + bx2. + + b2nxn = b,2y + b22y2 +... + bnîyn...(1), bnixl + bnx2 +... + bnx, =4by +by +... + bnYn where the system bik is skew, so that bik = - bki, bii = z........................ (2). The rows of coefficients on the right coincide with the columns on the left. Let B= bik I= bki I, so that B is a skew determinant, and let Bik be the system of first minors. Solving the system of equations (1) we get Yi = c1ilx + Ci2X2 +... +- Cinn k = dkly +- dk2y +... + dknyn. The coefficient of xk in yi is given by BCik = Bil blk + Bi2b2k +... +- Binbnk. If s = Bibki + Bi2bk2 +... + Binbkn then BCik + s = 2Bikbkk. 198 THEORY OF DETERMINANTS [CHAP. XIV. Now s = B or 0 according as i is or is not equal to k, thus 2Bikz 2Biz -B Cik- B ' C- B In the same way d 2Bikz 2Biz- B B -, BThus Cik = dki, and we may write yi = Ci1X1 + Ci2X2 +... + CinXn Xi = Cliy + C2iy2 +... + Cniyn. Substitute for x,... x from the second of these systems in the first and equate coefficients of Yk and yi on both sides, thls Ci2 + Ci22 +...+ Cin2 = ' Ci Ckl + Ci2Ck2 +... + CinCkn = O If we substitute from the first system in the second, we get C?, + C2 +..+ + C, = 1 CliClk + C2iC2k +... + CniCnk = ' Whence we see at once that x12 + x 22... + n2= Y2+ 2+... y2 and thus the coefficients Cik are those of an orthogonal substitution. 20. By the preceding article we are able to express the n2 coefficients of an orthogonal transformation by means of the in (n - 1) quantities bl2, bl3... b,, b2,, ~,, b23... b2,n bn-in ) by forming a skew determinant with these, the elements of whose leading diagonal are equal to z; and without loss of generality we may put z= 1. For the case n= 2, let B= 1, X =1+X2; -X, i 19-20] APPLICATIONS TO BILINEAR AND QUADRATIC FORMS 199 the system of first minors is 1, X -X, 1. Hence the coefficients of a binary orthogonal transformation are 1 -X2 2X 1 + X2 1 + X2' -2X 1 - X 1 + X2 1 +2 *X For a ternary orthogonal transformation B= 1, v, -/L =-1 +2 + 2 + 2; -, 1, X, -X, 1 the system of first minors is 1 + v X/, - + Xv, - v + XV, 1 +/2, X + ^, ~A + Xv, -X + /v, 1 + v. Hence the coefficients of the ternary orthogonal transformation are 1 + X --- v2 v- Xp - Vh -~ + Xv B ' B ' B -v+ Xi 1 + /2- X2 2 2 X + V 2 -— r ---,/-'2B B ' B ' B 2/A + - L 1 + v2 - X2_ -2 2 -g - 2- -— / — B ' B ' B If we write X=cosf tan iO,,J = cos g tan r0, v = cos h tan 10, where cos2f+ cos2 g + cos2 h = 1, and therefore B = sec2 0 we get Rodrigues' formulae. 200 THEORY OF DETERMINANTS [CHAP. XIV. For the quaternary orthogonal transformation B= 1, a, b, c -a, 1, h, -g -b -h, 1, f -c, g,-f, 1 Then B= 1 + a2 + b2 + c2+f2 + 2 + h2 + 02, where 0 = aj + bg + ch. And the system of first minors is B = 1 +f2 + g2 + h2, B = a +fO - bh + cg, B21 = - a-fO + cg- bh, B22= 1 +f2 + b2 + c2 B = - b - cf - gO + ah, B = - h + fg - ab - cO, B41 = - c - bf - ag- h, B, = g +fh + bO - ca, BI = b + gO- cf + ah, B4 = c + h - ag + bf, B23= h +fg+ c- ab, B24 = - g + hf - ac - b, B33 = 1+ g2 + c +a2, B34= f h + aO- bc, B4= - f + gh- bc- a, B44= 1+ h2 + a2 + b2. Thus the coefficients of the quaternary orthogonal transformation are Bc,, = 1 - 2+f2 -a2+ -2 _ b2 h2 -, Bc, = 2 (a +fO - bh + cg), Bc, = 2 (b + gO- cf + ah), Bc14 = 2 (c +hO - ag + bf), &c. 21. The square of the determinant of an orthogonal substitution is unity, for Cik 12 dik 1 where dik = CiCki + C2ic2k +... + CniCnk, i.e. dik =, d = 1; Cik 12= 1, or [cik =e, where e means + 1. 22. If Cik is the complement of cik in C, then Cik -e. Cik 20-24] APPLICATIONS TO BILINEAR AND QUADRATIC FORMS 201 For we have the system of equations Cn1Clk +... + CniCnCk = ClkCck +... + CnkCnk = 1 CCinck... + CnnCnk =0. Multiply these equations by Ci,, Ci... Cin and add, the coefficient of cik is e, the others vanish; thus Cik = -Cik, 23. Any minor of the system cik is equal to its complementary minor. For Cn... Cp =- ep- cp+lp+l... CP+fin.............................. CPl0... Cpp Cnp+il *- Cnn by vI. 6. But Cl... Clp -= p Cl *... cip Cpl... C*pp 4jpl.. - Cpp by the theorem just proved. Hence e ll.. Clp = Cp+lp+i... Cp+ln Cpi... Cpp Cnp+fl.. Cfll 24. If A() = an,, B() = I b, be two determinants of orthogonal substitutions of order n, then the determinant P (X, f/) = Xann+, + Iubîn is not altered by interchanging X and fi. For the symbolical expression for P (X, u) is P (x, u) = (XA + /uB)n =A ( A (~B( + ), as in VI. 7. And as there proved P(X,.)=A()'. BA ( + ( B (1> A (n> 202 THEORY OF DETERMINANTS [CHAP. XIV. Or, if A () = 1 =B<), we have by Art. 22, P (X, p) = | Xbik + gaik =P (/, X). From this we see, that if from the coefficients of an orthogonal substitution of order n we subtract the corresponding coefficients of another orthogonal substitution of the same order, the determinant formed with these differences vanishes if n is odd. 25. If we take n quadrics in n variables we may conveniently represent them by the system of equations u = aijkjxk (i, j, k 1, 2... n). With the coefficients aik we can form a cubic determinant of order n which will be an invariant of the system of quadrics u1... un. Zehfuss has pointed out that for three ternary quadrics this gives Aronhold's invariant, while the auxiliary expressions he gives for its calculation are the cubic minors of the second order. For the two binary quadrics ax2 + 2bxy + cy2 a'x2 + 2b'xy + c'y2, it is the harmonic invariant aa'- 2bb' + cc'. The general theorem is that for n quantics of order p in n variables the determinant of class (p + 1), which can be formed with their coefficients, is an invariant of the system. By allowing all the quantics to become identical we get an invariant of a single quantic when it is of even order. CHAPTER XV. DETERMINANTS OF FUNCTIONS OF THE SAME VARIABLE. 1. IF y,, Y2... yn are functions of a variable x, and if (k) d ckyi Yi -dxk the determinant yly2... yY2 =,... yn yl (), y2(1)... yn(1)....................... (n-1) (n-l) (n-1) Yi1,Y2... y.j is called the determinant of the functions Yi, y2... Y, and is denoted by D (yi, y2... n). 2. If y is any function of x, and we multiply the above determinant by y, O, O...O =yn, y(),,0 0. y(2), 2y(1), y...0 y(n-1), (n- 1) y(n-2), (n - 1)y(-)... y combining the columns of D with the rows of the latter, we obtain D (yyi, Y2... yyn) = y'-D (yi, y2... yn). In particular if we put yy, = 1 in the determinant on the left, all the elements in the first column vanish, except the first, which 204 THEORY OF DETERMINANTS [CHAP. XV. is unity, and the determinant reduces to the determinant of the n - 1 functions d (y)=-P y 2 Y ) Y) d (yl\= D (yi, y,) dx\y y... dx\yj y~ If therefore we put D(y,, Y)= Y!, then D (yl, Y y. ) = -- D (y2', y3... y). 3. If the functions y,...yn are connected by any linear relation Clyl + C2y2 +.. + C~yn = 0, it is plain by differentiating this n-1 times, and eliminating i... cn between the original and these n-1 new equations, that we get: D(yl,... yn)= 0. Conversely if the determinant of the functions y... yn vanishes, then they are connected by a linear equation with constant coefficients. We shall prove this by induction; we shall assume that if the determinant of n- 1 functions vanishes, these functions are linearly connected, and we shall shew that the same is true for n functions. If y, does not vanish, which would be equivalent to a linear relation among the functions, it follows from the preceding article that since D(y/, Y2.. yn)=O, we mrnst also have D (y2/ y/... yn) = 0. tLence by hypothesis the n- 1 functions y'... y' are linearly connected, t we have C2y/ + c3Y3 +... + Cnn' = 0. Dividing by y, we get d Cl / Y d^ = 0, dx \y dx \y. + d or integr ti+ig cy + c2y 2+... + cyn = 0. 2-5] FUNCTIONS OF THE SAME VARIABLE 205 Thus if the theorem is true for n - 1 functions, it is true for n, but it is clearly true for two functions, and hence generally. 4. From the formula D (yi, y... yn)= —2D(y, Y3... y), Yi it follows that D (yl, y2, y3)= - D(y, y3) Yi D(yi, y2, yn)= D (y, yn). The same formula also gives D (y2, y,... y,') = y, D D (y,', y,3), D (y2", Y4)... D (y2, yn,)) y2 Combining these formulae, we obtain the equation D (y, Y2...y,) y] n) {D (y, y, Y3), D (y,, Y2 y4)... D(yl, y,, yn)} By repeated application of this method we obtain the theorem:If Ui, u2... Um, V), v2... Vn be functions of x, and if Wi = D (u1, U... um, Vi) (i =, 2... n), D(W,, W...w,,) then D (ul, u2... m, v,....n) = {D (Wi, W,. Wn ID U i, U2... ~ tM/Jîl5. A special case of this theorem is D (y,... yk-1, Yk+... yn, Yk, Y) D {D (y..k-1.. y + y, yk ), D (1i... Yak-1, Yk~+.* Yn, y)} D (y... yk-1, yk+... Yn) which we mav write in the form D(y... yn, y)D(y... yk-i, yk+i - yn) _ d D(y, yi... ykyk+... yn) D (yi... yn) D (yi... y.) dx D (y... yn) 206 THEORY OF DETERMINANTS [CHAP. XV. Assuming now that the finctions yi... yn are independent, let us write Zk = ( I)n+k -D (Y.. Yak-1, Yk+i.- yn) D (y,... yn) p (y) (_ ])n D(y, y... n) D (y,...y.) P (y, Zk)=(-1)k-1 D (y, Yi..Y. k-1, yk+~i.* y) D (y,... yn) then the above equation can be written ZkP (Y)= - P (y, zk). 6. The determinant Y, y2... yn Yi(1), y (1)... yn(1) y (n-2)>, y(n-2)... y(n-2) (k), Y2(k)... (k) vanishes if k < n - 1, but if k =n -1 its value is D (y,, y,... yn). Expanding it according to the elements of the last row we get the system of equations y~iz + y2z2 +.. + ynzn = 0 y/i(1)Z + y2() z2 +... + yn(1)Z =0................................................. (A ). y,(n-2) z + y (n-2)z2 +... + yn(n-2)zn = O y (n-1) Z + y2(n- )Z... + yn (n-1)- =1 If we write Spq = yl(P)z,(q) + y2(P)Z2(q) +... yn(P)Zn(q), we can write these more briefly Soo = 0, S10o = 0. Sn-2,0 = 0, s 0 = 1. Now we have dSk-_, o dxS = Sk, o + Sk —i,, dks8o d-~Lk Sk, o + ksk~-, i + sCk-2, +. + So, k 5-7] FUNCTIONS OF THE SAME VARIABLE 207 If k < n - 1, it follows from these equations that Sa = 0, if a+ +3 <n-1. If k =n - 1, it follows since (1-l)r = 1 -r + r2-... + (- 1)r = 0 that Sn-_l,, s8-2,... are alternately equal to + 1 and -1. Hence we get the following theorem: The expression Spq is equal to zero when p + q <n - 1, and equal to (- 1)q when p + q =n-1. If k = n, we conclude in the same way that n,0 o = - Sn-i,i= 87n-2,2 = *.. = (- 1)) 0, n. 7. Among the relations just established we have zlyl + z2y2 +.. + ZnYn = 0 Z1(l)y + Z2(1)y +..+ Zn(1)y1 = 0............................................................(B ). l(n-2)yl + Z2(1-2)y +... + zn(-2) Y 0 Z (n-1)yl Z(y + z((n2 +. n(-)yn = (- l)n-1 If D (zi, z2... zn) vanished, it would follow, since S00 = 0, S01 = O * * So,n-2 = 0, that Son-1 would also vanish, while its value is (- 1)"-'. Thus the functions zi, z2... Zn are not linearly connected with each other. Comparing the systems (A) and (B) it appears that the relation between y,... yn and z1... zn is a reciprocal one, if we neglect the sign when n is even. Frorn each relation between these systems we deduce a new one by interchanging yI) y2 y** Yn, Z) Z... Zn with ( -1)-z, (- l1n-z2... (-1)-zn, yI, y2... yn Thus from the equation zk = (- )n+k D (Yi... Yk-, Yk+1... Yn) D(yl, y... yn) we deduce Yk =- k- (z1... Zk —, Zk+I.. * n) /D (z1 2 z... z.) In consequence of this we shall call z,... Zn the conjugates of y... yYn 208 THEORY OF DETERMINANTS [CHAP. XV. 8. If we form the product by rows of the two following determinants yl... yk, yk+l Y n ~..................,........, o....... (k-1) (k-1) (k-i) (k-L1) Yi' ~ Yk, Yk+l 'Yn y,(k).. y (k) ( (k) Y' YkS+l Yn I... 0, O... O................................. o... 1, O... O Z1... Zk Xk+i *** Zn z (n-k-i)... Zkn- (n-k-i) (n-k-). (n-k-i) the first of which is D (yl... yn), the second D (Zk+... xzn), we get Y... Yk, Skoo... k, n-k-i....................................... Y/1 yk, 8kc,o... lck,n-k-i yl( n-). yk("-l), Sn-i1,o...* n-1,n-kIn this determinant the elements common to the first k rows and last n - k columns ail vanish, whence it reduces to yl... Yk S, o... Sk, n-k-1,,.,,.................... y (k)... Yk (k) S,... Sn-l, n-k-1 Yi ( —1)... Yk (k —1) Sn-i, o... Sen-1, n-k-i The first of these = D (yl... yk); in the second all the elements to the left of the second diagonal vanish, whence its value is (n-k) (kn-k+l) (- l-) Sn-i, o n-2,.. Sk,n-k-i -1. Thus we have D (yi... yn) D (Zk+i... Zn) = D (yl... yk). If k = O, we have D (y... y,)D(z.. z)=1. 8-10] FUNCTIONS OF THE SAME VARIABLE 209 9. From this last equation we get p((Y) ( D (y, y... yn) D (y, y,... yn) = (- 1)n (y, i... yn) D (z,, z,2,n).) Or P(y)=(- l) y, y,... yn 1, O... o y(l), Yi(1)... yn (), z... z,, y (n), yi(. y(n 0, Z1'. z., z. (n2-1) = (- 1)n y, Soo00 So... So,n-1 y (1), Sio, Sn... Sllny (n), Sno Sn *... Sn, n-i Similarly we should get P ()=(- 1) z,z(... z (n Soo, ) oi..* - So........................ Sn-1, o f Sn-1,.. Sn-1, n 10. These determinants occur in the theory of linear differential equations. Thus, consider the equation ay + aly(l) +... + ay() = O where the quantities ao, ai... a,t do not contain y. If y,... yn are? particular integrals, we have the n equations aoyi + ayi(l +... + anyi(n = (1, 2.. n), and by eliminating the a's we get y, y)... y(n> =0, i................ yn, 2yn(1)." Yyn(n) or D (y, i... yn)=O. S. D. 14 210 THEORY OF DETERMINANTS [CHAP. XV. an S y i * (1).. - 2), yi( yi >.(.). yl(-1) _ _ _................................................ an 'Y1.1 yYl(1)... y. (-2) yn (n) (1)... Y, (n-i) d ___ i.e. 1d x 0-1. log D (y, y...2 yn)= a, D (y, y2. yn) =exp. (_- _a_ d 11. Though not immediately connected with the subject of the present chapter we shall give Hesse's solution of Jacobi's differential equation. This equation is -Aldl1 + A2d + A3 (fd - rd) = 0, where Ai = ail1 + ai2?r + ai3 (i = 1, 2, 3). We can write the equation in the form of the determinant,, v, 1 =0. dr, dr, O Ai, A2, A3 Now let -=, = '-; the equation becomes z z x, y, z =0. x' - z'x, zy' - yz', O Al, A2, A3 Multiply the first row by z' and add it to the second, this divides by z, and we get x, y, z =0. ', y', z' A1, A2, A3 Now let us multiply this equation by al, 1, 7y a2, f, 72, a3, 13, 73 and let pi = aix + /iy + 7iz. 10-11] FUNCTIONS OF THE SAME VARIABLE 211 Also assume that Xipi = Alai + A2,fi + A3,i. Then Pi, p2, pî =0, dpl, dp2, dp3 Xlpl, x2p2, X3p3 i.e. dpl dp2 dp3 =0, or logpl, logp2, logp3 =C, Pl P2 P2 p 1, 1, 1 1, 1, 1 X1, X2, \X X1, X2, X3 or, as we may write it Pi.P2 -Ph =- Since we assumed that Alai + A2i3 + A3,yi = Xipi we have, by equating coefficients of x, y, z ac (a, - X) +!3lai2 + 71a13 = 0, ala2, + 81 (a22 - X) + yla23 = 0, ai a3 + la32 + 7i (a,,33 - X) = 0. Hence eliminating ai, fi, y7, we see that XI, X2, X3 are the roots of the equation ai - X, al2, a3 =0. a2, a2 - X, a23 a31, a32, a33 - 14-2 CHAPTER XVI. APPLICATIONS TO THE THEORY OF CONTINUED FRACTIONS. 1. THE application of the theory of determinants to continued fractions gives great facility in the discussion of these functions. As usual in English mathematical works we shall denote the continued fraction bb 1 -- b 3 3 +... b, a, b1 b2 b3 bn by a, + a2 + a, + + a, Such a fraction is called a descending continued fraction. In addition to these we shall discuss a less known form of continued fractions, which, however, is historically the older form of the two, namely, the ascending continued fraction b, + "2 + bl - -t. a, ai which, in an analogous manner, will be denoted by b+ b2 + + bn al a,2 a6 Our object is to establish a determinant expression for the convergents to these two forms. 1-3] THEORY OF CONTINUED FRACTIONS 213 2. If we write down the system of equations blx = alxl + x2 b2,z- = ta2x2 + x b3x2 = a3x3 + X4 we see that x1 bi X2 b2 Wa + - a2 + - Hence 1 is the continued fraction bi b, a, + a2 + 3. If we are to determine the nth convergent, i.e. the value of the fraction when we stop at bn, we must suppose that xoen and -n all succeeding x's vanish, whence we have the system of equations bIx = aixi + x2 0 = -b2xr + a22x + x3 0= - b3x2 + a3 + x4............................... 0 = - bnxn- + ancn. Solving this set of equations for xI we get: a, 1,0... s1= blx, 1, 0 -b2, a2, 1..., a, 1... 0,-b3, a3... 0-b, 3 * o.oo................................. 0, 0, 0... an_, 1 O, 0...bn, an 0, O,...-b, an Thus l bi a, 1... 0 0 a,,... 0, 0 W -b3, a3... O 0 -b2, a, 1... 0,0 b........................ 0, 0, 3 0, 0, 0... a, 0.............................. 0, 0...-b, an 0, 0, 0...a, 1 O, O, O...-bn, a 214 THEORY OF DETERMINANTS [CHAP. XVI. or Xi - Pn x qn say, where n=b a2,,, 0... 0, 0 -b3, a3, 1, 0... 0,0 0,-b4, 4, 1... 0, 0................................ 0, 0,,0... an-, 1 0, 0, 0,... -b, an = anpn-l + bnPn-2, if we expand (iv. 24) according to the elements of the last row and column. Similarly qn= a,, 1, 0, 0... 0, 0 -b2, a2, 1, 0... 0, 0 0,-b3, a3, 1... 0, 0 o..........o............oo........ 0, 0, 0,0... an-_, 1 0, 0, 0, 0...-bn, an = anqn-l + bnqn-2. dq", Since pn = b, -, we can write the convergent in the form bdal 4. The determinants of the form qn have been called continuants by Mr Muir. Since qn n = aqn- + bnqn-2, if Un is the number of terms in the continuant of order n Un -= -6n-1 + Un-2, an equation of differences which gives = A (1 + )rB (-21 \/5 ) Since u, = 1, 2u = 2, we have ( = {(1 + V/5)n+ - (1 - 5)n+ll 2-n+i <5. 3-6] THEORY OF CONTINUED FRACTIONS 215 It is easy to shew by the binomial theorem that this number is an integer. Prof. Sylvester obtains this number in the form of the series 1 + ( (n - 2) (n -3) +( - 3) (n-4) (n - 5) 1+( -14 ) +. + +'... 1.2 1.2.3 5. The value of the continuant qn is the same as that of the determinant q'= al, C1, 0... 0 d, a, c2... 0 O, ds, a,... 0 0, 0, 0... a, provided only cd=.+ - = - b+ (r = 1, 2... - 1). This is clear if we expand by IV. 24, according to the elements which stand in the last row and column. For then qn = an' n-i - dc-n-2 = anq'n-l + bnqn-2 where qg' = q, q2' = q2. Hence qn' = q,, the equation of differences being linear. Thus we can also write qn= a, -1, 0,... 2, a2, -1, 0... 0, b, a3, -1... 0, 0, b4, a4... 6. The value of the continued fraction is not altered if we replace b,., a,., br+l by kbr, ka., kb,.+l. For the quotient Pn is unaltered if we multiply numerator qn and denominator by any the same number. If we multiply bot. by k, the row...- br a, 1... 216 THEORY OF DETERMINANTS [CHAP. XVI. in the determinants equated to pn and q, in Art. 3 is replaced by.. - kbir, kar., k.. and by Art. 5, in place of the last k, we can write unity if we replace br,+ by kb,.+,. Since then we can write the continued faction b2 b2 b1, al + a2 + ' + a in the form k k2 k2 k2 b - b2 b, b a, al2 a ' a3 ' a_n- an k+ k+ k + + k' qn can be written in the form of the skew determinant k, a, 0, 0... -a, k, a2, 0.. 0,-a, k, a... O,,-a, k..........................,,,,.. where a, = A/ ), Thus the convergents to a continued fraction can always be represented by the quotient of two skew determinants. 7. In any determinant D we have d2D dD dD dD dD da,,dann daa dann da,, dau For D take the continuant qn (Art. 5), then d2D 1 dD dD 1 da, dann, b,' l dan n-i, da1 b dD dD da = bb3... b., d (- lI)-L Thus qnpn-i - qn-lipn (- 1)12 blb2... b. 8. In the case of the ascending continued fraction b, + b2 + a1 a2 6-8] THEORY OF CONTINUED FRACTIONS 217 it is clear that if the nth convergent be P-n, the scale of relation is Pn = nlpn-l + bn qu a anqn-i Hence qn = aia2... an. To determine pn we have the system of equations: pl..........= bi - a~pi + p2 = b2 - ap2 + P3 = b3 - an-lpn-2 -F pn-1 = bn- anpî,_- + pn = b,. The determinant of this system is unity, all the elements to the right of the leading diagonal vanishing;.'pn= 0,0... O, bl -a2, 1, 0... 0, b2 0,-a, 1.. 0, b3 ooo............o...... O, O, O... 1,- bn-l 0,0, 0... -a,, b,, Multiply all the columns except the last by - 1, and move the last column to the first place; the determinant is unchanged, thus p,= ib,-1, 0... 0, O b2, a, -1...0, O b3, 0, a... 0, O b,, 0, O... 0, an The nth convergent to the fraction is P, n ala2... an The number of terms in p,, is n. 218 THEORY OF DETERMINANTS [CHAP. XVI. 9. By means of these determinant expressions for the convergents we can transform an ascending continued fraction into a descending continued fraction. In the determinant pn of the preceding article multiply the rth row, beginning with the last, by br_-, and subtract from it the (r - l)st row multiplied by br, and do this for all the rows. The determinant is multiplied by the factor blb2... bni =- k-, say, and pn=ck bl, -1, 0 0, a2b +b2, -b.. 0, - a2b3, ab2+b3............................................. O, 0, 0...a2b,-3+bn-2, -bn-_3 0, 0 O. -an-_n, aan-1bn-2+bn-1, -bn-2, 0, 0... 0, - an,, anbn-i+bn Similarly, since qn= aia2... an = a, -1, 0... 0, 0 0, a2, -1... 0, 0 0, 0, a,... 0, 0................................ 0, 0, O... a_;,-1 0, 0, 0... 0, an qn= al, -1,... -alb2, a2bi + b2, - b1 0, -a2b3, a3b2+b3..................................... O, 0,.. - a,_-b,, anbn, + bn Now on inspection it is clear that these determinants are continuants, as defined in Art. 3, whose 2nd, 3rd... (n - l)st rows have been multiplied by b1, b2... b,_2 respectively; also dqfl da1, 9-10] THEORY OF CONTINUED FRACTIONS 219 Hence by Arts. 3 and 6 pn bi alba a2b b3 a-2 bn-3 bn- an-lbn-2 b qn ai - a2b, + b2 - a3b + b3 " anb,-2 + b,_n - ab,- + b,' which gives us a rule for transforming an ascending continued fraction into a descending continued fraction, the number of quotients in each being the same. 10. We can make immediate use of this theorem to deduce a formula of Euler's, by means of which a series can be converted into a continued fraction. Take the series S = A, - A,4 +... + (- I),-1 An = A1, 1, 0,...0 A2, 1, 1,...0 A3, 0, 1, 1...0, An, 0, 0... 1 as we see by subtracting from each row the one below it, beginning with the last, when the determinant reduces to its principal term. Multiplying each column after the first by - 1, we reduce the determinant to the continuant for an ascending continued fraction. Thus the above series is equal to: 1)n- l + A2 + An- + An 1 -1 ' -1' — ' and transforming this by the rule just obtained to a descending continued fraction = (- 2 Y-1AA3 An-2An 1 - A2 - Al + A - A, + An - An-_ A,1 A2 A1A, A,_2A, i + A1 - A -A + A - A_- - An' If the original series is 1 I2 A1 A2 A3' 220 THEORY OF DETERMINANTS [CHAP. XVI. we can obtain its form as a continued fraction by altering the continuant to S in accordance with Art. 6, when we get 1 A12 A92 A, + A2 - A1 + A3 - A2 + 11. Various generalisations of continued fractions have been devised by Jacobi and others. The following generalisation, due to Firstenau, is taken from a review of his memoir by Ginther. If x and y are any two real numbers, and we write 1, 1 $1 Xl X2 W3 x = bo +, - = b + -, 2 = ba +-. Yi Yi Yi Y1 -Y2 Y3 where a and b are the greatest integers contained in x and y, then on substituting we have:, b + a4 b 3 + -- __ _a b2 +_ b4 a3+ a4 a3+b4 3 + ----- a4 and b2 +, a!3 + __I ___~~a4 +,a, + — 63 b a2 +-la4 10-12] THEORY OF CONTINUED FRACTIONS 221 If now all that stands to the left of one of the vertical lines be called a first, second... convergent, and if we denote the numerators of x and y by Xp, Yp, while the denominator, which is clearly the same for both, is called Np, we shall have (Y, X, N)V+l= ap+l (Y, X, N)pb + b_+l (Y, X, N)p_, + (Y, X, N)p_2. Thus the equations have four instead of three terms, and we get YP= CLo, bl, 1, 0... O -1, ao, b, 1...0 0, -1, a2, b.3 ~ 0 0, 0, 0, O... ac Xp= bo 1, 0, 0...0 -1, ai, b2, 1...0 0, -1, a2, bo...0........................... 0, 0, O, 0... a Npa= b1, 1,, 0... 0 - 1, a, b2, 1...0 0, -1, a2, b4...0........................... 0, 0, 0, O... a Corresponding to the theorem of Art. 7 we have now Yp+l, Y, Yp -1. 1. Xp+1, aXpa Xp-l Np, Np, Np-, 12. If ordinary continued fractions be called fractions of the first class, those in Art. 11 may be called fractions of the second class. Fürstenau extends the idea still further, and summing up his results we may state them as follows: If we seek to determinne n quantities x1, x2... x, as fractions of the form, X~ Xn x- =-l j,2 N2 ' -^ * N 222 THEORY OF DETERMINANTS [CHAP. XVI. each such fraction can be written as a continued fraction of the (n- l)th class. The pth convergents to these continued fractions take the form X~1 X2 X pVl xp2__ pn Np' Np N' and if a11 ** C " n+a il... a,, -1, a * *... a2, n+l (Ja+l, 1 * a Bn+l, n+l are the quotients entering into the continued fractions, then Xpq = aipXp,-l, q a+ a'2,X-2, q ~ * * *a +,1+, p xp-n-1, q, Np =-a pN_ +al N2_ +... + an+1, p Ap-nThe quotients X and N are always connected by the equation Xpl xp-1,, xp-2, *. p-n, = (- ). Xp2 x _p-12, 2 p-2, 2... Ypn, 2,,,.....,......,.......,........... X*pn) p-1, n) Xp-2, n. Xp_-n, 7 Np, Np_, N_2... p-n The author also shews that the real roots of an equation of the nth order can be represented as periodic continued fractions of the (n - i)th class. CHAPTER XVII. APPLICATIONS TO GEOMETRY. 1. THE axes being rectangular let the co-ordinates of the angular points of a triangle ABC be (xi, y y) (2x, Y2) (x3, y,). Then if A is the area of the triangle it is plain from the figure that Y B O N 1m X = trap. BN. - trap. BL - trap. CL = 2 (Y2 + Y3) (-2 - x3 2 + (Y2 2- YI)( ) - (1) Y3 + yl) (x1- x3), or 2A = y3x2- Y23 + x3y1 - xly3 + x1y2- x2y = 1, 1,> = 1, x1., yi Xl, X2, X3 1, 2, Y2 yl, /2, Y3 1, y3, 3 If the axes were oblique this would have to be multiplied by the sine of the angle between the axes. Thus 2A\=sin(XY) 1, 1, 1 i 12, 2C3 X yl, Y2, Y3 224 THEORY OF DETERMINANTS [CHAP. XVII. where (XY) is the angle between the axes. This form, however, is not often used, and unless the fact is specially mentioned the axes are supposed to be rectangular. If we multiply the first row by x1 and subtract it from the second, then the first row byr y, and subtract it from the third, we get 2/=- x2-x1, x2-x1 y2- Yi, Y3 - Yi It must be noticed that the area of a triangle changes sign if we alter the cyclical order of the letters. Thus ABC and ACB are equal triangles, whose areas are opposite in sign; ABC and BCA are equal in magnitude and agree in sign. 2. Let the co-ordinates of the angular points of a tetrahedron ABCD be (x1, y, zl)... (x4, y4, z4). Let V be its volume. Let A be the area of the triangle BCD, and let the equation of its plane be (x - x2) cos a + (y - y2) cos f + (Z - Z2) cos 7 =0. The projection of the triangle BCD on the plane of xy is A cos y, and the co-ordinates of its angular points are (x2, y2) (3, y3) (x4, Y4); thus, by Art. 1, 2A cosey = x- ax2, x4- x2 yY- Y2, Y4-Y2 Similarly we get 2A cos i = z -z, z4- 2 2 cos a = y - y2, /4 - y2 XS3- X2, X4 - 2 Z 3- Z2, Z4 - Z2 If p is the perpendicular from A on the plane BCD, - p = (x1 - x2) cos a + (yl - /2) cos f/ + (zi - Z) cos 7. Hence - 6V= - 2Ap = 2A cos a (, - 2) + 2A cos f (y, - y2) + 2A cos 7 (z - 2) =(xI1- ) y 3-Y2, Y4 - 2 +(Y/1-y2) z3-Z2, Z4-Z2 Z3 - Z2, z4 - Z2 X3 2- 2, 24 - 2 (Z- 3-) X 2, X4-X2 Ya — Y2, Y4 - Y2 1-3] APPLICATIONS TO GEOMETRY 225 _= X- X2 X3 - X4 - X2 = - 1 X, 1, 1, 1 Yl-Y2, Y3-Y2, Y4-Y22 X -X2, 0, X3- X2, X4- X2 Z1-, 3- -, 4 - 2, 4 - Z2 -,,y2 -, Y4 - y2 Z1 —Z2, O, Z3 - Z2 Z4-Z2 If in this last determinant we multiply the first row by x2, Y2, z2 and add it to the second, third and fourth rows respectively, we obtain 6V= 1,, 1, 1, 1 Xi, X2, X3, X4 Yi, Y2, Y3, y4 Z1, Z2, Z3, Z4 3. If the tetrahedron be referred to oblique axes through the same origin, and if the cosines of the angles these make with the rectangular axes be given by the scheme X Y Z XYZ 1l 12 13 y ml qM2 m3 Z ni n2 n3 = XI, + Y2 + ZI3, &c.; whence 1, 1, 1, 1, 1, 1 1, 1, 0, 0, 0 X1, XW, x2, 4, X1, X2, X3, X4 0, Ml, m i, yl, Y2, Y3, y4 Y1, Y2, Y., Y4 0, 12, m2, n2 12, Z2, Z3, Z4 Z1,, 2, 3, Z4 0, 13, m3, n3 Now let D= li, mi, ni 12, 2 2 2 13, m3, n-3 then remembering that I2+ m2 + nm 2 = 1, 1112 + f mlm + 1n,2 = cos X Y, &c., S. D. 15 226 TIIEORY OF DETERMINANTS [CHAP. XVII. we have D2= 1, cos XY, cos XZ cos YX, 1, cos YZ. cos ZX, cos ZY, 1 This determinant is usually called the square of the sine of the solid angle contained by the oblique axis, in analogy with the determinant sin2XYI 1, cosXY cos YX, 1 in a plane. Thus D2= sin2 (X YZ). And in oblique co-ordinates 6 V= 1, 1, 1, 1 sin (XYZ). X1, X 3, X2, X 4 YI, y2, Yi, Y, Z1, Z2, z3, z4 4. From the determinant expressions in Arts. 1 and 2 we can at once write down a number of geometrical relations. If the distances x be measured along a straight line from a fixed point, we see that 1, xi = (xCk - i) = (k) 1, xk is the distance between the two points marked k and i. The determinant 1, X1, 1, x1 1, X2,, X2 1, X3, 1, 3 1, x4, 1, X4 vanishes identically, because it has several columns alike. Expanding it by IV. 5 according to products of minors from the first two and last two columns, we get (12) (34) + (13) (42) + (14) (23) = 0. 3-6] APPLICATIONS TO GEOMETRY 227 Or, if we call the points A, B, C, D, this is the well-known relation between the segments formed by four collinear points AB. CD + AC. DB + AD. BC = 0. If we expand the vanishing determinant 1Xi, ix y,, 1, x, yi \ (=1, 2... 6) according to minors from the first three and last three columns, we get no geometrical relation, the terms cancelling each other in pairs. But if we expand the determinant 1 1x, Yi, i,, 1,, yi, zi, =0 (=1, 2...8) according to the products of minors from the first and last four columns we get an identical relation of thirty-five terms between the volumes of the tetrahedra formed by eight points. 5. Again, for five points, 1, 1, 1, 1, I =0. 1, 1, 1, 1, 1 X1, X2, X3, X4, X5 Y2, Y2, Y3,, Y4, y5 Z1, Z2, 23, Z4, Z5 If v1= volume of tetrahedron (2345) and we expand the deterrrinant according to the elements of the first row, by IV. 10, we get v1 + V2 + V3 + v4 + v5 = 0. 6. By the theorem VI. 20, 1, 1, 1 1, 1, = 1, 1, 1 1, 1 Xl, y2, Y3 1, 2, V3 Y1, l2 2, 32, X3 Y2, Y2>2, Y23 5, '2, 773 2, 'q1 2 '3, Y2, Y3 + 21, 1, XI 1, 1, 1 1,1, 1, Xi 1 e1e, X1> el > X2, X3 Yi, -2, 3 7i, y2 Y3 yl i Ç2, y, Y3 Or if the two sets of three points be called ABC, DEF, ABC x DEF = ADE x FBC + AEF x DBC + AFD x BCE is a relation between triangles. 15-2 228 THEORY OF DETERMINANTS [CHAP. XVII. The product of the two determinants 1, 1, 1, 1 1, 1, 1, 1 Yl, X2 X3, YX4 el, 12,,3, V4 y Y12 YS, Y4 7, '2) z3, '4 Zl, Z2, 23, Z4 X1, X2, `3 4 can be represented either as a sum of four terms 1, 1, 1, 1, 1, 1, +..., x1, 2',,) C l3, e1 2, e3, 4, X4 Y,1, Y/2 Y3,?71i 72, '73, '> 4, /4 | Z1, Z2, Z3, 1 2 3>;4 Z4 or as the sum of six terms 1, 1, 1, 1, 1, 1, 1 +....!1 Zy, y i1,,2 V2 3, V4, 03, Z4 Or calling the two sets of points ABCD, EFGH, we have the identical relations between the volumes of tetrahedra: ABCD x EFGH = ABCE x FGHD - ABCF x GHED + ABCG x HEFD - ABCH x FGED ABCD x EFGH = ABEF x GHCD + ABGH x EFCD + ABEG x HFCD + ABHF x EGCD + ABEH x FGCD + ABFG x EHCD. Application of Alternate Numbers in Geonetr-y. 7. In applying alternate numbers to geometry, a number stands for a point in a flat space whose dimensions are one less than the number of units. To begin with a plane, the units el, e., e. stand for the vertices of a fundamental triangle ABC. Any other number P = xe, + ye, + ze, 6-7] APPLICATIONS TO GEOMETRY 229 stands for some point in the plane of the triangle. It is generally convenient to assume that x +y + z- 1, so that x, y, z may be taken to mean the ratios of the triangles PBC, PCA, PAB to the triangle ABC, though this is not necessary. If P and Q are two points, then mP + nQ m + n is a point in the line PQ, dividing PQ in the ratio n: m. Thus ~ (P + Q) is the middle point, and P - Q the point at infinity of PQ. Similar definitions hold for a space of three dimensions. Four points ABCD being taken and represented by the units el, e2, e3, e4 any other point in the space is represented by P = xel + ye2 + ze3 + we4, where if we choose we may write x + y + z + w= 1, x being the ratio of the tetrahedron PBCD to ABCD. And so on for a space of any number of dimensions. Then a binary product e.eS is a unit length measured on the line joining the points er, es or the distance between the points e., es. A ternary product e.eet is a unit area measured on the plane of the points er, es, et, or the area of the triangle formed by the points er, es, et. And so on. In a space of two dimensions the product of three points is the area of the triangle they form referred to the fundamental triangle. Now if P = xiel + y1e2 + zle3, Q = x2e +... R = xsel+.. PQR= x,1 Y1, Z1 ele2e3. X2, y2, Z2 X3 YV, Z3 230 THEORY OF DETERMINANTS [CHAP. XVII. And ee2e3 = ABC = A, the area of the fundamental triangle, so that in areal co-ordinates PQR= xi, y,, z A. X2, 2 y2 Z2 X3, Y3, Z3 Similarly in a flat space of three dimensions if ele2e3e4 = V is the volume of the fundamental tetrahedron, the volume of the tetrahedron formed by four points is PQRS= x,, y,, W, W V. x2, Y2, Z2, tU2 X3, y3, Z3, W3 x4, Y4 Z4, W4 Similar definitions may be stated with reference to flat spaces of more than three dimensions. The assumption which has been made throughout the present work, that the product of all the units of a system is unity, receives here its justification and explanation. For, geoinetrically speaking, the product of the units is the measure of the fundamental figure of the space considered, which is our unit of measure. In a plane, for example, it is the area of the triangle of reference, in ordinary space of three dimensions the volume of the tetrahedron of reference. It is no part of the plan of the present treatise to develop the geometrical applications of alternate numbers; for these we must refer to the memoirs and works of Grassmann and Schlegel. Angles between straight lines. Solid angles. Spherical figures. 8. With rectangular axes let l, ml, il, Xi, /Ui, Vi 12, m2, n2 \ 2, V2 7-9] APPLICATIONS TO GEOMETRY 231 be the direction cosines of two sets of straight lines, then cos (ik) = liXk + mik 4- I iVk is the cosine of the angle between the ith line of the first and kth of the second system; and, by compounding the two arrays, we get the determinant I cos (ilc). Hence by v. 3, if there are two sets of four straight lines we get cos (11)... cos (14) = 0...............(i). cos (41)... cos (44) If there are two sets of three straight lines a, b, c; f, g, h, cos af, cos ag, cos ah = il, mn, ni X1, p1, vl cos bf, cos bg, cos bh 12, mn2, n2 X', /2, V2 cos cf, cos cg, cos ch 13, m3, n X3 3, 3, = sin (abc) sin (fgh)............(ii). If there are only two straight lines in each set I cos(), os (12) i= i, m k, i +.. i cos (21), cos (22) 1 12, 12 | 2, /-2 Now if n, v be the directions of the shortest distances between the lines of each pair, and 0, e the angles between the pairs, i 1!,' = =n sin 0 cos (nz), &c. cos (11), cos (12) = sin 0 sin 5 cos (n)......(iii). cos (21), cos (22) 9. If in the relation (i) of Art. 8 the two sets of straight lines coincide with one set of straight lines a, b, c, d. we have I, cos (ab), cos (ac), cos (,di) = O. cos (ba), 1, cos (bc), cos (!)d cos (ca), cos (cb), 1, cos (cd) cos (da), cos (db), cos (dc) i 232 THEORY 0F DETERMINANTS [CHAP. XVII. This is the identical relation between the mutual inclination of four straight lines in space, or also the relation between the sides and diagonals of a spherical quadrilateral. If we write - cos (AB) for cos (ab), or what comes to the same thing change the signs of the elements in the leading diagonal, it becomes the identical relation between the cosines of the dihedral angles of a tetrahedron formed by four planes A, B, C, D perpendicular to the lines a, b, c, d. 10. If the two straight lines marked 1 coincide with two straight lines u, v; while those marked 2, 3, 4 coincide with a set of oblique axes x, y, z, cos 6Uv, cos ux, cos uy, cos uz =0, cos Xv, 1, cos Xy, cos xz cos yv, cos yx, 1, cos yz cos zv, cos zX, cos zy, i which gives the cosine of the angle between two straight lines u, v, referred to a set of oblique axes x, y, z, in terms of their direction cosines. 11. As another example of the use of the same formula, let ABC, A'B'C' be two spherical triangles, 0, 0' the centres of the small circles circumscribing them. For our two sets of straight lines take the lines joining the centre to O'ABC, OA'B'C'. Then if 00'= b, and R, R' are the radii of the circumscribing circles, we get cos b, cos R', cos R', cos R' = O. cos R, cos (AA'), cos (AB'), cos (AC') cos R, cos (BA'), cos (BB'), cos (BC') cos R, cos (CA'), cos (CB'), cos (CC') We can write this cosqsin(A BC) sin (A'B'C')=- cosRcosR' O, 1... 1 1,cos(AA')...cos(AC')..cos(C.A')......cos (C) 1, cos (CA')... cos (CC') 9-13] APPLICATIONS TO GEOMETRY 233 If the angle at which the small circles eut is F cos = cos R cos R' - sin R sin R' cos; and the above formula can be written (1 - tan R tan R' cos r) sin (ABC) sin (A'B'C') =- 0, 1... 1 1, cos (AA')... cos (AC')................................ 1, cos (CA')... cos (CC') If the two systems coincide f == 7r, and we get sec2 R, 1, 1, 1 =0, 1, 1, cos c, cos b 1, cos c, 1, cos a 1, cos b, cos a, 1 a, b, c being the sides of the spherical triangle. 12. Similar relations can be developed in the same way for a plane. In a plane we can shew that for two sets of three straight lines cos (11), cos (12), cos (13) =0, cos (21), cos (22), cos (23) cos (31), cos (32), cos (33) and then deduce 1, cos C, cos B = 0, cos (xy), cos (xa), cos (xb) = 0, cos C, 1, cos A cos (ay), 1, cos (ab) cos B, cos A, 1 cos (by), cos (ba), 1 similar to the equations in Arts. 9 and 10. 13. Next, let us compound two arrays 1, l1, ml, ni 1, -\ X1, -i, - Vi.......,o,,.......................... 1,, n, mp, np 1,-Xp,-L,-p. We get the determinant 1 - cos (ik) \ = 2 sin2 (ik). 234 THEORY OF DETERMINANTS [CHAP. XVII. Hence, by v. 3, for two sets of five straight lines sin2 (11)... sin(15) = 0..................(i). o..................... sin2 2 (51)... sin2 (55) For two sets of four straight lines a, b, c, d; a', b', c', d', 16 sin2 (aa)...si(n2(ad') - 1, li, l m,, ni x 1 X, i,, vi........................... (i = 1, 2, 3, 4)............(ii). sin21 (da')...sin2 (dd') Expanding the determinants on the right according to the elements of their first column, our determinant = {sin (bcd) + sin (cad) + sin (abd)- sin (abc)} x {sin (b'c'd') + sin (c'a'd') + sin (a'b'd') - sin (a'b'c')}. For two sets of three straight lines, our determinant is 1 - cos(ll)... 1 - cos (13) j........................................ 1 -cos (31)... 1-cos (33) or 1, 0... O 1, -1,... -1 1, 1 -cos(11)... 1-cos(13) 1, - cos(11)... -cos (13) 1................................. 1,.............................. 1, 1 - cos (31)... 1 - cos (33) 1, - cos (31)... - cos (33) This is equal to the sum of the products of determinants of the third order taken from the two arrays. Omitting the term 11, m1, ni -X1, -/, - v = - cos (11)... - cos (1 3) t2, m2, n I -\2, - /2, -^2............................................... j 12, m2, 2 2, - V2 13,I3, m3 n3 -13, - - cos (31)... -cos(33) we get 0, 1... I 1 Il,m|l,X, ul + 1,,nl 1l,X, v 1, cos (11)... cos(13) + l,m,n!1l,v. ~...........,......... 1, cos (31)... cos (33) If the straight lines be called a, b, c; a', b', c', and NV, V, N3 13-14] APPLICATIONS TO GEOMETRY 235 are the directions of the shortest distances between bc, ca, ab, we have 1, 1, m = sin (bc) cos (N7I) + sin (ca) cos (Nz) + sin (ab) cos (Nz), I1, X, I = sin(b'c')cos(Nl'z)+sin(c'a')cos(N2'z)+sin(a'b')cos(N,'z), and similarly for the other determinants. In particular, if a b c lie in one plane, and a'b'c' in another, the normals to the two planes being N, N', the value of the determinant is {sin(bc)+sin(ca)+ sin(ab)} Isin(b'c') + sin (c'a') + sin(a'b')} cos (N1T'), viz. this = 0, 1... i 1, cos (aa')... cos (ac').... (iii). 1, cos (ca')... cos (cc') For two sets of two straight lines we deduce in the same way, if R, r are the directions of the external bisectors between them, 0, 1, 1 ab a'b' O l 1 = - s4 sin n. cos (Rr). 1, cos (11), cos (12) 2 n 1, cos (21), cos (22) 14. If we compound the arrays il, ml, ni, 1, 0 X1, /i, vl, 0, 1 li, mi, ni, 1, 0 Xi, Ui, vi, 0, 1 0O, 0, 0, 0,1 O,, 1,0, we get the determinant cos (11)... cos (l),1 cos (il)... cos (ii), 1 1... 1, O Hence for two sets of five straight lines cos (11)...cos (15), 1 =0. cos (51)... cos (55), 1 I i... i 236 THEORY OF DETERMINANTS [CHAP. XVII. For two sets of four lines cos (11)...cos (14), 1 =- 1,,,, n x 1, \I, v,........................... cos (41)... cos (44), 1 I... 1, 0 and so on. But these are not new theorems. In the first, for example, if we expand by iv. 24, according to products of elements in the last row and column, each terni vanishes by Art. 8. On Systems of Straight Lines. 15. If w-p y-q_z- r s a cos cos cos be the equations of a straight line, then a = cos a, b = cos e, c = cos y, q r, g= r p, h= p q cos /, cos cos y, cosa cos a, cos f are called the co-ordinates of the line. It is plain that af+ bg ch = 0. 16. If the constants belonging to two straight lines be denoted by the suffixes 1 and 2, the equation of a plane through the second line, parallel to the first, is x-p2, y-q., z-r2 =0. cos al, coS,31, COSy1 COS a2, cos/32, cos y2 If d be the shortest distance between the two straight lines, and 0 the angle between them, it follows that d sin 0 = pi - p, q -, ri - r2 cos ac, cos ]3l, cos 7y COS a2, COS /32, cos 72 = p, ql, ri + p2, q2, r cos al,,, cos i y, cs 71 oss a,2 cos 2, cos 72 COS a2, cos /3, cos 7 cos ~c, cos /3, COS = a2fi + b2g, + c2h, + alf2 + bg2 + ch2. 14-17] APPLICATIONS TO GEOMETRY 237 If the expression on the right vanishes, then either d = O, i.e. the two straight lines intersect, or sin = 0, when they are parallel, and hence also meet. It is convenient to have a name for the expression on the right. If a unit force acted in one of the lines its moment about the other would be d sin 0, i.e. in terms of the co-ordinates of the lines alf2 + bq2 + lh2 + a2,f + b2gl + c2hl. Hence we shall call this the moment of the two straight lines. If two straight lines meet their moment vanishes. 17. Let us take two systems of straight lines whose coordinates are a, bi, c1, fi, gi, hi fi, gi, hi, ai', b,', c/' Then if m,, denotes the moment of the line r of the first and s of the second system, by compounding the two arrays we get the determinant mik 1 Hence for two sets of seven straight lines n11... m17 = O,...... m7 an identical relation between the mutual moments of two sets of seven straight lines. If the two systems coincide 0, qt.. ml7 = 0. ml,,0.. m27.................. 71,^ 72... 0 For two sets of six straight lines mil... m6 = I ai, bci, fi, g, hi I........... x fi g', i ',h', ai', bi',... 6). M61..- m766 If one of the sets of six straight lines-say the first-is met by a common transversal whose co-ordinates are a, b, c, f, g, h, we have for each of the straight lines of that system afi + bgi + chi +fai + gbi + hci = 0. 238 THEORY OF DETERMINANTS [CHAP. XVII. Thus the first of the determinants on the right vanishes, and MnI,... im6 = 0.o.......... rnG6... m66 is the relation between the mutual moments of two sets of six straight lines, one set of which is met by a common transversal. If the two sets coincide we get the identity for a system of six lines met by a common transversal. 18. If the moments of a system of forces about one set of seven lines be ml, mn... m,, and about a second set ni, n2... 27, we can establish an identity among the moments involved. For if any force P of the system act in a line whose co-ordinates are a, b, c,f, g, h, we have ml = SP {afi + bg1 + chi +fa, + gbi + hcl =f/iPa + g#ZPb + h iPc + a, SPf + b, 1Pg + c, Ph, and six other equations for mn... rn7. Hence eliminating ZPa, PPb... PPh, we get ni,, ai, bl, c,, f, gi, h = O, n7, a7, b7, C7,7, 97, h7 and a similar equation for the other system. Hence each of the determinants O, mi, ai, bl, cI, fi, gl, hl ni,, fi', gl, hl', a/, b/, c'............................................................. 0, 7, a7, b7, C7, f7, g7, h7 n7, 0, f> 7/, h7/, a/, b7, C7 1, 0,0, 0, 0 0,0, 0, 0,0,, 1,0,,,, 0, vanishes. Forming their product we get n11... rM17, nl = 0................. m71... r77, n7 mi... n7, O 17-19] APPLICATIONS TO GEOMETRY 239 Tetrahedra and Triangles. 19. Let there be two systems of points in space whose coordinates referred to rectangular axes are (xi, Yi, zi), (i, i, i,). Let us compound the two arrays X1,,, 1, 1,0 -2,-2,- 2, 0, 1.......*...,.................................... x, yi, zi, 1, 0 - 2- 2i, - 2-, 0, 1 0, 0,,, 0 0, 0,, 1, 0, we obtain the determinant Cil... Ci, 1 Cil... Cii 1 I... 1 where Cs =- 2Xrs - 2yrs - 2z-.rs. To the rth row add the last multiplied by xr2 + y2 + zr2, and to the sth column add the last multiplied by e,2 + r2 + s2, the determinant is unaltered and its elements are now drs = X,2 + yr2 + z1. - 2x,.s - 2yr.s - 2zs2 + s + 2 + s2 = (Wr - eS)2 + (yr - +s)2 + (Zr -?)2, i.e. dr, is the square of the distance between the rth point of the first and sth point of the second system. We have then the determinant dl... dli, 1 oooo........... dil... diitt, 1 If i = 5 the determinant vanishes, hence dll... dl5, 1 =0........................(i)............... ds5... d55, 1 1... 1 is the identical relation which subsists between the lines joining two sets of five points in space. If the two systems coincide di = 0, and the determinant, which is then symmetrical, gives the relation between the lines joining five points in space. The relation in this form is due to Cayley. 240 THEORY OF DETERMINANTS [CHAP. XVII. If i = 4, d11...d14,l = X1,y1,1,l,0 x 2, - 221 0, 1 di,..-. d-4, 1 X, yl, Z, 1, 0 x - 2~1, - 2Vl, 2 0, 1....,........................................................ d41... d44, 1 x4, y 4, 1, 0 -- 2 4, - 2V4, -2 4, 0, 1...,, o, o, 0o, O, o, O, 1, 0 = 288 VV'............................................(ii), where Y, V' are the volumes of the tetrahedra formed by the two sets of four points. If the two sets coincide in a single tetrahedron, for which a, a'; b, b'; c, c' are pairs of opposite edges, 288 V = (, c', b'2, a', 1 C'2, 0, c2, b2, 1 b'2, c2, O,,2, 1 a'2, b2, a2, 0, 1 1, 1, 1, 1, 0 If i= 3, we have dl... d3, i =-4ixy,ly,,1-4xz,l, 11,l-4yz,l -,lz,............ooo d3... d33, 1 1... 1, 0 all the other determinants on the right vanishing identically. Now if A, A' be the areas of the triangles formed by the two sets of three points, (1, m, n), (X,,a, v) the direction cosines of the normals to their planes, x, y, 1 1 = twice projection of A on plane xy = 2An, and similarly for the others; hence if j is the angle between the planes of the triangles dn... d, = - 16AA' cos b............... (iii). d31... d33, 1 1... 1,0 Lastly, if i = 2, dll, d2,, = x1, 1, O - 2, 1 +... d2l, d22, 1 x, 1, 0 - 22, 0, 1., 1,1 0 0, 1 O, 1, 0 = 2 (x, -,2) (e - t2) + 2 (yi - Y2) (, - -r2) + 2 (zl - z2) (1 - ), 19-20] APPLICATIONS TO GEOMETRY 241 the other terms vanishing. Now if a, b be the lengths of the lines joining the points of the first and second systems and 0 the angle between then, X1-X2 ~1.-2+... +... cos0. a b Hence d1l, dl2, 1 = 2ab cos.....................(iv). d2l, d22 1 1, 1,0 20. If in case (iii) of Art. 19 we allow the two sets of three points to coincide with the vertices of a single triangle whose sides are a, b, c, -16A2 = 0, c2, b2, 1 2, 0, a2, 1 b2, a2, 0, 1 1, 1, 1,0 Multiply each column by abc, then - 16A a4b4c4= O, abc, abc, abc abc3, O, albc, abc ab3c, a3bc, O, abc abc, abc, abc, O Divide the first, second, and third rows and columns by bc, ca, ab respectively, then -16A2= 0, c, b, a c, 0, a, b b, a, 0, c a, b, c, O = a, b, c, 0 b, a, O, c c, O, a, b O, c, b, a by an interchange of columns. If in the first expression for - 16A2 we divide the second and.. D. 16 242 THEORY OF DETERMINANTS [CHAP. XVII. third columns by a2, and then multiply the first and last rows by a2, we get: -16A2= 0, C2, b2, a2 c20, O, 1, b2, 1, 0, 1 a2, 1, 1, 0 21. If in case (ii) of Art. 19 one of the sets of four pointssay the first-lies in a plane, then V= 0, and dll... d4, 1 = 0. d41.. d44, 1 1... 1 If one of the sets in case (iii) lies in a straight line the corresponding triangle vanishes; hence dl... dl3, 1 =0. ds3... d33, 1 1... 1 By allowing the second system to coincide with the first we get the identical relations between the lines joining four coplanar and three collinear points. 22. In the identical relation dal... ds, 1 =.0 d51... d,5, 1 1... d between the squares of the lines joining two sets of five points, let the fifth point of the first system be the centre of the sphere circumscribing the tetrahedron forrned by the first four points of the second system, and the point 5 of the second system the centre of the sphere circumscribing the first four points of the first system. Then d(5 = d25 = d35 = d5 = R2 d51 = d52 3 = d= 4 = R'2. 20-23] APPLICATIONS TO GEOMETRY 243 Also, if ( be the angle at which the two circumscribing spheres intersect, d55 = R2 + R'2 + 2RR' cos ~. Hence with an interchange of rows and columns dl... d, 1, R2 = 0................ o..... 41... d44, 1, R2 1... 1,0, 1 R'... R', 1d, 55 Multiply the fifth column by R2 and subtract it from the last, and the fifth row by R'2 and subtract it from the last, then dl... d14, 1, 0 =0............................... d... d44, 1, 0 1... 1, 0, 1 0... 0, 1, 2RR'cos~ Or, resolving according to the elements of the last row and column, we have by Art. 19 (ii) 57 6 VR V'R' cos b = d1... d14 d41... d44 We see from this that so long as the circumscribing spheres remain fixed the tetrahedra can turn about in them without altering the value of the determinant on the right. The determinant vanishes if the circumscribing spheres of the two systems cut orthogonally. This relation is due to Siebeck. 23. If in Art. 22 we allow the two tetrahedra to coincide we get, since ( = wr, 16(6VR)2=- 0, a'2, b'2, c2 a'2,, c2, b2 b'2,, O, a2 c', b2, a2, 0 16-2 244 THEORY OF DETERMINANTS [CHAP. XVII. Multiply the second, third and fourth rows and columns by a2, b2, c2 respectively, then 16 (6VR)2 a4bc444 - 0, (aa')2, (bb')2, (cc')2 (aa' )2, 0, a2b-2C, a2b2C2 (bb')2, a2b2c2,, ac2bc2 (cc')2, a2b2c2, a2b2c2, O Divide the second, third and fourth rows by (abc)2, then multiply the first column by the same quantity, 16 (6VR)2 — 0, (aa')2, (bb')2, (CC')2 (aa')2,, 1, 1 (bb')2, 1, 0, 1 (cc)2, 1 1, 0 Now if we write aa'= kx, bb'= ky, cc' = kz, then if A is the area of the triangle, whose sides are x, y, z, we have by Art. 20, (6VR)2 = 1k42, 6 VR = k2À. This triangle, whose sides are proportional to the square roots of the products of pairs of opposite sides of the tetrahedron, has many interesting relations to the tetrahedron. It is sometimes called the conjugate triangle. Formuloe relating to the Ellipsoid. 24. Let (xi, yi, zi) and (,i, mo, ri) be two sets cf points on the ellipsoid X2 y2 z2 4- - + 1 a2 b2 c2 Then, if dr. denote the square of the distance between the rth and sth points of the two systems and D,.s the square of the parallel semidiameter, we have ars = ds _ 1 _rts _ yr7s Zs\ -ar = a-2 - b- -~. 23-25] APPLICATIONS TO GEOMETRY 245 Hence, if we compound the two arrays, xl yl 1Z I f 21 21 a' b'c' a' b ' c xi yi Zi 2i 27i 2i 2 a' b' c' a ' b ' c we get as in the preceding articles:For two sets of five points situated on the ellipsoid, a1... a,5 = 0. a5... a55 For two sets of four points forming two tetrahedra of volumes a,.. a14 576VV'........... ce a 2b2c2 a4l... a44 Similar formule cari be established for an ellipse in a plane. If the ellipsoid become a sphere, a = b = c = R, and since all diameters are equal, we can replace a,, by d,. Thus dl... dl5 =0 d5i... d55 is an identical relation between two sets of five points on a sphere. This relation is due to Cayley. The second relation in this case reduces to the result of Art. 22, when the two tetrahedra have the same circumscribing sphere. 25. If the points (xi, yi, zi) (i, i, Oi) are not situated on the ellipsoid, then since dr _ (xr - ) (r-s)2 (r - ) )2 ars = + -— 2 - C 2 Y 1 2 9 2,reS 2yl-qOs 2z1 Cs es2 +2 1s+ - + + Z'- 2 - b+ cS a' a' b zc' 2a 2yb' 2c + + v/ ~Y a-2 c b2 c a c2 246 THEORY OF DETERMINANTS [CHAP. XVII. if we compound the two arrays whose rth rows are xi2 yi Zi2 xi Yi i 1 a2 b2 ' c a' b' c' 1 a-1 ++, _, _ a ' b ' c ' a2 b2 2 we get the identical relation (v. 3) ai...a16 =0 a6n... a66 for any two systems of six points in space, and an... a5 - xi+ y+ zxi x yi, z a2 b2 c2 a', >,, c, a 5 x I, b 1 c a2 b2 C2 (=1, 2...5), for any two systems of five points. If in the latter equation all the points of the first system lie on the ellipsoid (-p)2 y - 2) (z - r)2 n2 a ---i + b7 + - ^m' we have X2 y' z2 2px 2qy 2rz p2 q2 r2 a + + +- + +- + = ma2 b2 C2 a2 b2 c2 a'b c2 satisfied for each point of the system. Hence we see by eliminating -2p -2q -2r p2 q 2 r2 a ' b ' c ' a b2 c2 between these five equations, that the first determinant on the right vanishes. Hence all... a15 = 0, a51... a55 if the five points of one of the systems lie on an ellipsoid similar 25-26] APPLICATIONS TO GEOMETRY 247 and similarly situated to the given one. If the ellipsoid reduce to a sphere, we get d1... d6 = 0, d61... d66 an identical and homogeneous relation between the lines joining two sets of six points. And d,... d15 = 0 d5l... d55 for five points situated on a sphere. 26. In like manner, if for the same systems of points as in the last article we compound the arrays, Yi 1,0 2 2 1 0 2, 1 a' b 'c ) a ' b ' c iyi zi -2i 2i O a'' c a b ' c ' 0, 0, O,,, 0, O, O, 1,0 we get the determinant cil... Cli, i Cii... Cii, 1 1... 1 2xrCs 2vr/s 2Zr., where c.s= a2 - 2 C- Multiply the last column by s2 s2 s2 a2 b2 c2 and add it to the sth column, and the last row by Xr2 Yr2 Zr?2 a2 b + c2 248 THEORY OF DETERMINANTS [CHAP. XVII. and add it to the rth row, then the element at the intersection of the rth row and sth column is (Xr - )2 +Yr 7)2+ (Z s\2 And hence (v. 3), a5... a55, 1... 1 1 is an identical relation between any two sets of five points in space. If the ellipsoid becomes a sphere we regain Cayley's relation (Art. 19, i). For i = 4, we have an... al, 1 288VV'.............. a2b2c2 a41... a44, 1 1... 1 V, V' being the volumes of the tetrahedra formed by each set of four points. 27. The polar plane of a point P (Xr, yY, Zr) with respect to the ellipsoid is XXr yy ZZr_ v+ -:=1. a2 b2 C2 The distance of a point Q (es, %s, Çs) froin this plane is i - + i' - +. p2 a4 b4 c4 If (Q, P) and q denote like quantities for the point Q, (P, Q) (Q, P) I x s _ Yr7.s zrs p q a2 b2 c2 This function has been called by Faure the index of the two 26-29] APPLICATIONS TO GEOMETRY 249 points P and Q; let it be denoted by Irs. Then, by compounding the arrays whose ith rows are xi yi zi -e -i -i - 1 a' b'c a b' c we obtain Il...I =-0 I51... I55 Iî... |14 36VV'........... 1 a2 b2c 41..' 44 28. It may be remarked that these space relations connected with an ellipsoid are not really more general than those connected with a sphere. For they may be deduced from the latter by applying to the whole configuration the homogeneous pure strain which changes the sphere x2 + y2 + z2 R2 to the ellipsoid x2 y2 Z2 a2 b2 c Formulce relating to Systems of Spheres. 29. If r, s be the radii of two spheres, q> the angle at which they intersect, and d the distance between their centres, then d =r2 + s2 + 2rs cos b. The function 2rs cos = d2 - r2 - S2 is of importance in the study of the mutual relations of spheres; it is called the power of the two spheres. We shall denote it by p,.s If one of the spheres, say s, becomes a point, the limit of 2rs cos q is d2 - r2, i.e. the square of the tangent from the point to the sphere, or what is known as the power of the sphere at the point, or the power of the point with respect to the sphere. 250 THEORY OF DETERMINANTS [CHAP. XVII. If both spheres reduce to points the limit of 2rs cos ( is d2, the square of the distance between the points. If one of the spheres becomes a plane, and p is its distance from the centre of the other, cos =. r If the second sphere become a point, and p is its distance from the plane, the limit of r cos b is p. 30. Let (xi, yi, Zi) and (ek, %k, k) be the co-ordinates of the centres of two spheres of radii ri and pk, then if pik is their mutual power pik = d2 - ri2 - pk2 = Xi2 + yi2 + Zi2- ri2 - i- 2yik- 22ik + k2 + rk2 + + k2 - pk2 Hence, compounding the two arrays x1, yl z1, 1, LlS2 yl2 + z12 2-rl2 Xi, Yi, zi, 1, x2 + yi2 + i2 - ri2 and -2e, - 21,- 2, + 2 + 1,2 + 2 - pl2, 1 o,,oo.o.,o.....,o..................... - 2i-, 2i, '2i, + i2 + - i2 - Pi2, 1, we see by v. 3 that for two systems of six spheres 11P... p1 = 0.....(............... (i). P61... P66 If cos ik is the cosine of the angle at which two spheres cut, we can also write this coso 1|=0 (i, =1, 2... 6). For two systems, each of five spheres, Pl... l5.....................................(ii) P51... p65 = Ix, Yz, 1, x2+ y2 +z22 r -2t, -2, -2 2 + 2 + p2 1. If the five spheres of one of the systems-say the first-have a common radical centre, taking this for origin we should have x2 + y2 + z2 - r2 = C2, 29-31] APPLICATIONS TO GEOMETRY 251 where c is the same for all the five spheres. Hence, in the first determinant on the right of (ii), the fourth and fifth columns are proportionals and the determinant vanishes. Thus pl...p =.......................(iii) P51....55 when the five spheres of one system have a common radical centre. If the five spheres of the first system reduce to points (iii) is the condition that they should lie on a sphere. If both systems reduce to points we regain Cayley's condition, that the five points of one system should lie on the same sphere. 31. But if neither of the determinants on the right of (ii) vanish, expand the first determinant with regard to the elements of the last column. Then pi = xi2 + yi2 + 2z? - ri2 is the power of the origin (i.e. any point) with regard to the ith sphere of the first system. Then if we write 1, 2, 3, 4, 5 for the centres of the five spheres, and denote by v, = (2345), v2 = (3451), &c., the volumes of the tetrahedra formed by the points in brackets, and if accents denote similar quantities for the second determinant, we have in place of (ii) pik | = 288 (vlp, + v2p2 +... + v5p5) (Vl'p' +... + v5'p5/) (i,/=1, 2...5). Now describe about the origin a sphere of radius r, cutting the spheres ri... r5 at angles J... 5. We have, since (Art. 5) v1 + v2 +... + v5 = 0 identically, v1p+. +..+ v5p5 = vi (pi1-r2) +... + V5 (p5-r2) = 2r (vlri cos ~l +... + v r5 cos q5), and p being a similar sphere for the second system, Pik = 288pr 2viri cos fivi'pi cos i/ (i, k = 1... 5). 252 THEORY OF DETERMINANTS [CHAP. XVII. Thus r 2viri cos fi is independent of the particular sphere r; let this be the orthotomic sphere of the first four, and let R denote its radius; then this sum reduces to 2v5r5R cos (rR), and the second factor, in like manner, becomes 2v5'p5R' cos (p5R'). Hence ll *... p15 = 1152v5v5'r5p5RR' cos (r5R) cos (psR'). P1.... P55 32. For the fifth sphere of each system in this last equation take the orthotomic sphere of the first four spheres in the other system. Then in the determinant on the left all the elements in the last row and column vanish except p55, and ps = 2RR' cos (RR'). Hence we obtain pln.. P4 2RR' cos (RR')= 1152v5v,'R2R'2 cos2 (RR'), P41... P44 or dividing out the common factors and writing V, V' for v,, v,', we get for two sets of four spheres pi... pi4 =576 VV'RR' cos (RR'). p41 *'. 44 If the spheres reduce to points we regain Siebeck's formula (Art. 22). The determinant on the left vanishes if the orthotomic spheres of the two systems of spheres eut orthogonally. 33. To determine the meaning of the determinant |Pi } (i, k = 1, 2, 3). In the determinant of Art. 32, let the fourth sphere of each system be the plane determined by the centres of the first three spheres 31-34] APPLICATIONS TO GEOMETRY 253 of the other system, then if A, A' be the areas of the triangles formed by the centres, b the angle between their planes, V V lim. = 3A cos /, lim.-= 3A' cos ô. "4 r"4 Also if the radical axis of the spheres of the first system meet the plane of centres of the second system in P, whose power with reference to the spheres is p, and P', p' denote like quantities for the other system, 2RR' cos (RR') = PP'2 -p -p'. Hence p,... pi3 = 16AA' cos b (PP/2 -p p'). PS..3 P33 34. In the relations d,... di, 1 =0, ds5i... d55, 1 I... 1 d,...d,4, 1 =- 288VV', d4l... d44, 1 1... 1 of Art. 19, let us suppose the sets of points to be the centres of our spheres. Then if we multiply the last column by p2 and subtract it from the ith column, and the last row by rk2 and subtract it from the kth row, we get the relations p...pi5, 1 =0, p51... P55, 1 1... 1 l... P4, 1 =-288VV',............... P41.. p44, 1 1... 1 254 THEORY OF DETERMINANTS [CHAP. XVII. connecting the mutual powers of two sets of five spheres and two sets of four spheres. 35. Another element connected with two spheres is the length of their common tangent. For two spheres of radii r, s the distance between whose centres is d and which eut at an angle Q5, the square of the length of the common tangent is given by t = d2 - (r - s)2 = rs cos2 M. If one sphere reduce to a point, t is the power of that point with respect to the other sphere. If both spheres reduce to points, t is the square of the distance between them. 36. Using the same notation as in Art. 30, if tik is the square of the tangent common to the two spheres tik = (Xi - k)2 + (yi - 1k)2 + (i - k)2- (ri - pk)2 = iL + y2+ Zi2- r-2-2xik - Yik- 2Zik +2ripk +k2~+k2+ k2-pk2 Hence, compounding the two arrays X1, Yi,, r, 1, XI2 + yl~ + Z12 - ri2 ooooo.. oooo............................ xi, Yi, zi, ri,, 1, x2 + yi2 + z2 - ri2 0, 0, O, 0,0, 1 - 2e, - 21, - 2, 2pî, fî2 +?1î2 + iC2 _ p2 1 - 2. - 2, - 2.i 2p. ei + qi2 + i2 pi2 1 0, 0, 0, 0, `1, 0, we get for two systems of six spheres the identity tll... t16, 1 = 61... 1 For two systems of five spheres we should get t,... t,5, I = 576 (vri +... + v,5,) (v,'pi +...+ V,'p),............ t5i.. t55, 1 1... 1 using the notation of Art. 31. 34-38] APPLICATIONS TO GEOMETRY 255 If t5 is the angle at which the plane of similitude of the first four spheres of the first system cuts each of these spheres, and (r5t5) the angle at which it cuts the fifth sphere, and similarly for the second system, we can reduce this to the form til....5l >, 1* COS (rt) COS (p)75 t t = -576v5r5v5'p5 (1 - (os 1 - cos.T t51... t55, 1 1...1 Hence the determinant vanishes if one of the systems of five spheres has a common plane of similitude. For two sets of four spheres, after some reduction we can prove that tll..l = 288v t cos 1............ cos t Cos T/ t41... t44, 1 1... 1 where b is the angle between the planes of similitude of the two systems, and t, T the angles at which they cut their sets of spheres. 37. By compounding the arrays whose ith rows are Xi, yi, zi, ri, 1, xi2 + y.2 + Zi2-r r and - 2i, - 2, - 2Ci, 2pi qii2 + 2 + 2 - p2, 1, we get the homogeneous relation between the sets of tangents common to two sets of seven spheres tll... t7 =0. t7l... t77 38. We may make use of this last relation to solve tlie problem: Determine the equation of the sphere having with five given spheres tangents of the same length. Let the equations of the five given spheres be s1=0 o...... =. 256 THEORY OF DETERMINANTS [CHAP. XVII. Take these for the first five of each set of spheres in Art. 38, let the sixth sphere be the onè required, and the seventh a point on the sixth. Then we shall have t67=, t7i = Si, t6i =, and the equation is O, t12, t3, tI4, t15, 1, s1 =. t21, () t23, t24) t25, 1, S2 oooooo....................... t51, t52, t53, t54, O, 1, S5 1, 1, 1, 1, 1, 0, 0 S1, S,, 83, 4, S5, 0, 0 This is apparently of the fourth order, but by means of the sixth row and column we can get rid of the terms of the second degree in the seventh row and column. 39. All the equations of this section relating to spheres are capable of numerous and varied applications, some of these will be found in the examples, and others in the memoirs of Bauer, Darboux and Frobenius. EXAMPLES. PROVE the following relations: 1 -5. 1. (b + c)2, ab, ac = 2abc (a + b + c)3, ab, (c + a)2, bc ac, bc, (a+ b)2 (b + c)2, c2, b2 2 (bc + ca + ab)3. c2, (c+a)2, a2 b2 a2, (a + b)2 2. 1, 1, 1 =0, tanA, tan B, tanC sin2A, sin2B, sin 2C if A, B, C are the angles of a triangle. 3. 1, x, (a + ) + ) = 0,,?J (a+ y) $(C+ y) 1, %, (a + ) +/( + ). a -c - ffa -c - c/ -3 a. if tan- ( c) + tan-l / ( — + tan-l / (c =) 0. C \+/ C + y C + z 4. 1, COS a, cos(a+p), cos(as(a + ), cos(a~+y-+y+a) =0. cosa, 1, cos, cos ( + -), cos ( + y + + ) COS(a + /Y), COS, 1 ) COS y, COS (+) cos (a + /3 + y), cos (/ +7), cosy, 1, cos8 cos(a/3++y+ê), cos(+y-+8), cos (y+8), cos ~ 1 5. a+b+c+d, a-b-c+d, a-b+c-d a-b-c+d, a+b+c+d, a+b-c-d a-b+c-d, a+b-c-d, a+b+c+d 16 (bcd + acd + abd + abc). s. D. 17 258 THEORY OF DETERMINANTS [EX. 6. If a, b, c are the sides of a triangle of area A, 2s=a +b +c, then (b + )2, ab, ac, ca =-16sa(a2'2+b4r2+c2r3), ab, (c + a)2, bc, b ac, bc, (a + b)2, c a, b, c rl, r2, r. being the radii of the escribed circles. If the elements in the principal diagonal are (b - c)2, &c., the other elements being as before, the value of the determinant is - 16 - + b + c) S Vi' 9'2 3 (b +- c)2, ab, ac, a - 16sA(a, + br +cr-) ab, (c + a)2, bc b ac, bc, (a b)2, c 1, 1, 1! (b + c)2, ab, ac, 1 6A - 20abcs. ab, (c + a), bc, 1 ac, bc, (a + b)2, 1 1, 1, 1 7. If S=al + a2 +... + an, A = S- a, prove the following theorems: x - A1, a2... ac = x (x- S)21, C, xC -A... a~n.............................. Al, a2... x- c^ x-aL, A2... A = {x + (n-2) S}(x - S)n. Al, x-a,... An A1, Ae... x-C a 8. The determinant a, b, b, b...... a, b a, a...... b, b, a, b...... a, a, a, b...... (the diagonal consisting of a and b alternately and each row being filled up with the other letter) is equal to (-y 1)- ( - 1) (a - )2. The determinant is supposed to have 2n rows. 6-13] EXAMPLES ON THE METHODS OF THE TEXT 259 9. If in a determinant all the minors of the second order are divisible by the same quantity p, then the minors of the umth order are divisible by pl2-l. 10. If in a determinant of the nth order there be a block of p by q elements all of which are divisible by a, the determinant is divisible by ca+-~. 11. Prove the theoremis: a, b, c, d,... = a~, a, a+b, a+ b+c, a+ b+ c+d,... a, 2a + b, 3C + 2b + c, 4a + 3b + 2c + d, a, 2ab+, 3a+2b+c, 4a+36b+2c+,... a, 3a + b, 6aC + b + c, 1Oa + 6b + 3 cl,... a, b, c, d... a= 1 -1. 221-2. 31-3... (n-. 1 ), a, a+b, a+2b+c, aa+ 3b+3c+d... a, 2a+b, 4a+4b+c, 8a+6 12b+ 6c d... a, 3a+b, 9a~+6b+c, 27a+27b+9c +d... where a, b, c, l... are any quantities whatever, and n is the order of the determinant. In the first determinant each row after the first is obtained from the preceding by the rule that the rth element of any row is the sum of the first r elements of the preceding row. In the second determinant the rth element of any row is the sum of the first r elenients of the preceding row multiplied respectively by the coefficients in the expansion of (1 + x)'-1. 12. If D - a, b, c, d... (n rows), - a, b, p q... - a, -b, c, r... -a, -b, - c,... then D= 2-1 abcd... The elements of the first row and leading diagonal are a, b, c, d...; in each column the elements below the leading diagonal are equal to the element in the first row but of opposite sign, the others are any whatever. 13. If D = cos na,, cos (z-1) a... cos a,, 1 cos nal, cos (n - 1) a1... cos ac, 1............................................. cos na, cos (n - 1) a... cos a,, 1 17-2 260 THEORY OF DETERMINANTS [EX. D, cosa0o, cosn-la,... cos a, 1 cos al, cos5-lai... cos a,, 1...................................... COSnan, coSn-la1... COS a,, 1 D2 = sin (n + 1) ao, sin nao... sin ao sin (n + 1) a1, sin na,... sin a...................................... sin (n + 1)a,,, sin na,,... sin a, then n (n- 1) n (ni+1) =D1 - 2, " = 2 sin ao sin a,... a,,. 14. If bi = (ai + aa +... + ai,) - aik, then b.l... b. n = l cl)... bnl... bnn nl... a2nn But if bi =- (a,, + ai2 +... + an) - 2ai b1..b b, =(n- ( 2)(-2)1-1 a,... a, bnl. -. bnn anl... an 15. Prove that every power of a symmetrical determinant may be expressed as a symmetrical determinant of the same order. 16. If for each element aik of a determinant A we write in turn aik + c, we get n2 new determinants. If these be taken as the elements of another determinant its value will be (Ac)n-l (c + S), where S is the sum of all the elements of A. 17. If (X, - abl) (X, - ab,)... (, - ab), prove that the value of the determinant Xa, aX2b, a3b2... ab2 alb3,, 2, b... a,,b3............... i............ alb,, abn,.ab,....X is u{~ Jl+ _ X. b }a++ (:-a~b?~b~ '[ 13-18] EXAMPLES ON THE METHODS OF THE TEXT 261 and the value of 0, al, a2... an b, X1, a2b... Caib b2, alb2, X2... a,,b2......................... f albl aabA ) is -U 1 lb, X - + +,b- b Xi,-ai bi XI - abitj 18. If = (x - 2al) ( - 2a2)... (x- 2a,,), prove the following theorems: (x-aj)a, 2! ', =... a + }, a 2,(x —a,)2, ca2 * - -2a' a12, a22, (x-a3)2....................................... 0, 1 1 1, 1. _,- cl 1,( -, (x -, a22, a2... d 1, a2, (x - c)2, 32... 1, a12, a22, (X-a3)2... a1a2, (X - a)2, a,23 *- CaL3, a2a3, (X- a-)2... O, a, C2, &3... -a- a2 C6l, (x - Ca)2, a2, a... - 2a a2, a1ia2, (x- a2)2, a2a3.. aC,, a,1a3, a2a, (x -a)2............................................. And if 1)= (x -a)2, a,... a2, b1, 1 a1, (x- a2)2... a-, b2, 1 a2, a2... (x- 1), b,, 1 1b, b2... b1, 1, 1... I 262 THEORY OF DETERMINANTS [EX. then xC-2u, '"- 2al + ' + 2l x- 2a, b, b1 x - c2al+ x - 2a 19. Prove that, if S-=x+ y +z+, u2 (/- )2 22 l y Z U ) ' +- _+- % y2 - -,2 x 2, (S y>)2, z2 u2, 2 y (S_-)2 0, 1, 1, 1, =S {x2(y+ +u) + 2(xz+zu) 1, (S-u)2, x2, 2, 2 (x ) + (2(x + + ) +(Z++ y) 1, 2, (S -)2, y2, z2 + 2-xyz + 2xzu + 2yXz + 2xyuZ 1, u2, 2, (S_ -)2, z2 - y3- z3 - u-3}. 1, a2^, 2, J2, (S - )2 20. If X = cn x dn x, &c. prove that sn x, sn3x, X = sn (y - ) sn (z - x) sn (x - y) sn(x + y + z) M sny, sn3y, Y snz, sn3z, Z where Mf= 1 - k2{sn2y sn2z + sn2z sn2x + sn x sn2y} + k (1 + k2)sn2xsn2ysn2z-k2snx snsnz (YZsnx + ZXsny + XYsnz). 21. If sn x cn x dn x = X, &c. prove that 1, sn2x, sn4x, =0, 1, sn2y, sn4y, Y 1, sn2z, sn4z, Z 1, sn2u, sn4u, U provided x + y + z + = 2pK + 2qiK', p, q being integers. 22. If Sijaj = + ai+7,.+j+7+ - aCi+h, j-a.i, j+l, then Si, S2.... SS1 2 S921î 22 ***- -;S2. -h;.......................... I 18-24] EXAMPLES ON THE METHODS OF THE TEXT 263 is the sum of all the minors of order k - of the determinant A _=; aik;; excepting always in such sum those determinants and their complements of order h which in their formation have two row or column suffixes congruent with regard to the modulus h. 23. If -- 0 1,, 1, 1... (nrows), 1,, x,, 0... 1, y, O, x, 0... 1, Oy, O, x... 1, 0, O, y,.0........................ where al] elements are zeros, with the exception of the border, and two lines of elements one on each side of the principal diagonal, prove that D2n =- xy D2-2- - It + y D2n+1 - xY D2?1)n + 2+ 2 (- y)" x+ y x+y and hence that Dx*- (- y)__, 1 +l_ ( + 1+ (2 ) (x ) (- x:y) 2_,1+ - (x + y)2 24. If Dn= c, a, c, c, c... ( (n rows), b, c, a, c, c... c, b, c, a, c... c, c, b, c, a... c, c, c, b, c... where all the elements are c with the exception of two lines, one on either side of the principal diagonal, prove that D(a — c) (c-b) Fin,-d -- c th value — + b -D. Find also the value ob- J Find also the value of Do,2. 264 THEORY OF DETERMINANTS [EX. 25. If D,,= 0, 1, 1, 1, 1... (n rows), 1, c, a, O, O... 1, b, c, a, O... 1, O, b, c, a... 1, 0, O, b, c. (where, with the exception of the border, the elements in the leading diagonal are c, in the lines on either side of it a and b, the rest are zero), then D,, - cDA,_ + abD,; 2 =- + a+b+b+c 2ab u2 - vn-2 a+b+ c U -v where u and v are the roots of the equation z2 -cz + ab= O. Hence shew that?- + vun nc (u'" + vi") (a + c) (a + b + +c) (u - v) 2abn ue1-l + v-' (- a)"- + (- b)?? a+b+c (z - v)2 (ab+c)2' 26. The value of the determinant u1, t2... ' it uibl ) ZG 1*`- Il2-1,n?,... Z,_...................... j Ui2 ZG3... i1 (i) If u,= a + (r-1) bis 2a + (Z- 1)b -— ^ (-^6g)?-l (ii) If XI='-, is (i- xit-) (iii) If,. = r2 is (- 1 (1 - {1)( + 1) ( -- 2), 1-....i....... { ('*+ )- '" 25-29] EXAMPLES ON THE METHODS OF THE TEXT 265 (iv) If 2,. = cos { + (r- 1) b} is [cos a - cos (a + nb)] - [cos ( - b) - cos {a + (n - 1) b}] 2 (1 - cos nb) (v) If zu.=sin {a + (r-1) b} we must change the cosines in the numerator of (iv) into sines. (vi) If u = x -1 + x+'-1 + x?+2n-1 +... ad inf., is (1 -x) —. 27. The solution of the partial differential equation Ds2... =0,,, D1... D9_lD2, D3... D where D dx. is u = n F(x2 - ox1, x3 - 2x... x- -1 ) the functions being arbitrary and the summation extending to all values of o being roots of the equation x- 1 = 0. 28. If in an orthosymmetrical determinant of order n (vII. 20), a k - _.._ _ (1 - qa) (1 - qa+). (1 - qa+;-2) ak = (1 -qY) (1 - qY+)... (1 -qy+-2) the value of the determinant is equal to /1 -a0-1 /_a+l i-2 1 _ Qa+b-2 \1-qYj 1- + J... _ q.+.L-2 multiplied by a fraction whose numerator is n (n- 1) n(n-1) (n-2) (- 1) 2 q 3 (1-q)-l (1 -g2)-2... (1 - qf-l) x (qY - qa)l- (qy+l - qa)l-2... (qy+l-2 _ q and denominator (1 — q) (1 - qY+l)2... (1 qY+f-2)lx (1 - qy+~-l)l-l (1 -Y+fT)-2... (1 - qy+2-:). 29. The value of the determinant D= 0,a 1 c a+, ai 1 +a3c... ( rows), (6. + 1, 0, 62 +,3.. a3 + Ca1, C, + 0.............. 266 THEORY OF DETERMINANTS [EX. the elements in the leading diagonal being zero, that in the ith row and jth column aic +aj, is given by (_ 1)Y D = 2baa,... a,, (1 - n -, (ai - a) 4 ai ai, J where i, k are all duads from ], 2... n. 30. The value of the cubic determinant of order n, such that Ca7k = ai + ai + Cak+ Ciii -- O, is given by (-i1)- D 1 eR.- (,- ac>) 3nala2... 6 ( cti at And if aC,,c = cos (ai + aj + Cai), aiii 0, ()- _ 1D = zc - 1( + a2 ) sin2 (ai - k) cos 3a% cos 3a2... cos 3a,, ces 33a cos 3a c where i, k are all duads from 1, 2... n. 31. If A = Iccl, B= Ibil are two determinants of orders n and cm respectively, we can form a new square array of (nrn)2 elements as follows. Repeat the array blk, n times in a row, and take n such rows, so that B is repeated like the squares on a chess-board. Then multiply each of the elements of that block which stands in the ith row and kth column by ca1i. The determinant of the resulting array is equal to A'"B1. Example: - b; B- a; 1P c, d aa, a/c, ba, b3 =A2B2. ay, a8, by, ba Ca, cp, cla, d/3 cy, c3, dy, d3 32. If a, b... 1; a, 3... X are any two sets of n quantities, and di = (C6i - a )r + (bi - P) +... + (i - 7 prove that dl... is =0, if s=( -1) + 3,........... csl.. Csi" 29-34] EXAMPLES ON THE METHODS OF THE TEXT 267 dl... d,, 1 i =O, ifs=n(r-1)+2................ 1... 1 1 33. {In this and the next five questions In(m-1) ( - 2 2)... ( -. + 1) ) ' i=nk= - 1. 2. 3 k -- ' The determinant mp, ~p+ l ' " m,, (mn + 1),, (m + 1)+i... ()n + 1)............................................................... (m + r' - 1),, (n, + - )+... ( + r - 1), (n + r)p, (In + )-1... (nt + ),, (rm + r + s)p, (+n( + * + S)1+1...* (M + r + S),i (m+rf+s+l),, (m+r+s+ l),+... (mi+r+s+l), ( + r+ s + t),, (m~ + r + t)p+... (m +r +s+ t), where u= p + r + t + 1 (the suffixes p, p + 1,... c of the rows are consecutive, but m, m + 1,..., +r, mn+ r+s,... + r + s + t form two groups of consecutive numbers), is equal to the product of the two fractions mnp (m + )p... (m + r'), (nm r + s),... (mt +. + s + t)p lPp (P + )p... (r + Ts) + (r + + + l)+,... (r + s + + (r + 1)( + (r 2).+.(r + t + l),.+ 34. The determinant p,* +.....P *... v +l........î m4+.s~.r....rp+S+V+ut (i + 1l), (ni + 1) +,... (n + 1 )+,,, (n13 + l ),+s+v.. (mi + 1),+s+v+~, (m + 2)p, (( + 92)p+l... (n + 2)+, (.? + +...- (nm + 2i)p+S+V+u.............................................................................. (m + r)p, + )+ (ni ), (Im + r))++~... (n + p+ ( + )r)++.. (,+ ),,++ where r = s ++ 1 (the suffixes p, p + 1,... p + s, p + s + v,... p + s + v + form two groups of consecutive numbers, while nm, n + 1,... m + r are consecutive), is equal to the product of the two fractions m,,> (_n + l)J)... (?n + r)p pp (p + 1),... (p + s)p (p + s + 'v)... (p + 8 + v + (,n - \) ( v- 9 l_ \ (m,- +?1 \- (v - 1)n-, V-i (v +' l)-l... (V+ - 1),?, 268 THEORY OF DETERMINANTS [EX. 35. Prove that x, Po, Pi... P.-o (X+1) (p + l)o, (p +l )1. ( +l)?-1 (x +2), (p+ 2)o, (p + 2),... (p +2)?_ (x + r)", (p + r)o, (p + r),... (p + r),._l vanishes if n<r, but is equal to (- 1)"n! if n = r. If nz>r the determinant reduces to a function of x of order n - r. 36. Prove that x Pl l... Pl -, ( x-P) (x + 1)", (p + 1)... (p + 1), (x + 2)", (p + 2)... (p +2),................................... (x + r)n, (p + r),... (p + r)r for all positive values of n less than r. 37. Prove that Po Pi.... P-i q =" ' A". (p + l)o, (p+ 1)... (p+ 1)-., (4 + 1)' (p + 2), (p + 2),... (p+ 2),._, (n+ 2)".................................................. (p + r)o, (p +- r),... (p + r),._l, (n + r)"' 38. Prove that the value of the determinant (m -p) m, n rn+~, q mp+2 t p+,, (m-p-p+l)(rn,+l),p, (n+l)(n,+l)p+l, (q+l)(t+l1)p+2, (t+l)(4i-+ 1)p+3... (m-p+ 2)(rn,+ 2),, (n + 2)(m + 2),+, (q.+2)(m + 2)1,+2, (t+ 2)(m + 2),+3... (mrnî-pt+ )(on + Ir)P, (n +r)(n + r)il~ (q +r)(n + r),+,2 (t + Jr)(m + r)p,,... is (m + 1)p...(m+? (On - p) (n - m p + 1) (m -p + 2)... (m -p + r), p1 (p3+ 1)p.. (p + r)p and so is independent of the quantities n, q, t... 39. If A= L ak [; B = bk \ are two determinants of order n, and f/(x) = { + x b,, prove that f(x)f(- x) = AB Hit - Iikv X2, where the quantities Hik, Ki, satisfy the equations H, K, + H1, K,. +... + I.,, + 1,, = 1, H, Kls + H2 L, +.. + HI n =7bs= 0. 35-41] EXAMPLES ON THE METHODS OF THE TEXT 269 40. With the same notation as in the preceding question, prove that if P (X, X) = ji la + )bi, |, then P (x, )=A /I1+ X, EH12... HL ff1, H 22+X... H IInl, Hqt,... P Sn + A AK9I, +.............. X...... K,,, x K2,... X Ài,, + ~ 41. If F (x) = aox' + alx'-1 +... + alex + a,, prove that a, 0 F(x) P= O, O., =.oo - --, ai0 a0 i;0, 0, 0... x,.......2 a0 0 O, 0... 0, Ya0 x, 0,x... 0, 0 ~a1~ a o O, O, 0...-1, +x ~a0 a I 0 x, 0, O... 0, i an 0oPrs + a Q2 a,3 xiO, 1... O, 0 -- xo, O, O.. 0, 1 If Pl.,>, Q,., be the coefficients of homologous eleinents in P and Q, ao P, x + a,, Q,.Q = 0. 270 THEORY OF DETERMINANTS [EX. Also, if to the elements of P we add the homologous elements of Q multiplied by y, the resulting determinant is equal to F () F (y) a aG,, 42. Prove the formula for the change of the independent variable in the determinant of n functions dy2 d2y3 d-12, dx cdix.. dX-1 dt(n+1) dy2 d2y3 cl'"-L y ~dxJ 2 yl dt2" dt-'t2 43. Let a,, a2, a3... be a series of n positive numbers, and let s, be the sunm of the divisors of r selected from the terms of this series, this sum being supposed to vanish for all values of r which have no divisors in the above series. Then if Dn = Sln-11 8' 2 S3 *.. SS - - S 818l-2 + S- n- 1 S, -82... s,_3 SlSî-3 3+ Sn-2 O, n -2, -s8... S -4 S1lSî-4 + 8_-3, O O n -- 3.. - n-5 iss1+2, 0, O, O... 2 the number of positive integral solutions of the equation aClXt + a2x2 + Ca3X +... = Di n! 44. If si. is the sum of all the divisors of r, then the determinant Sit-l Sn S1 82 S S3 ** su-3) sit-2 8o_ - î8, 81 ~ 82 83 *... 8._ 3 8_Sn Sn-2- Sb-l n - 1, 81 * —, 8 -. -3 n_-3- Snl-2 -- S-1 n S..-5î Z~-4 S?,-3 - 82 0 -, 8- 2.. Sn-î 51 -45 Sîo_4 --,_-, 0, O, - 3... Si-_, Sn_5...................................................... S2-83, O, 0,... 3, s s, — S, 0 O, 0,... 0, 2 is equal to (- 1)n! when n is of the form 2 (3k2 - k), but vanishes for other values of n. 41-49] EXAMPLES ON THE METHODS OF THE TEXT 271 45. Let (mn, n) denote the greatest commnon divisor of the integral numbers nm and n; anc let q/ (m) be the number of numbers prime to m and not surpassing m; the symmetrical determinant o,7, = S ~ (1, 1) (2, 2)... (,, m) is equal to O (1) S (2) ~ (3)... i (2). 46. If A is a skew determinant of order n in which the principal diagonal elements are equal to z, and A i, its system of first minors, prove that A-l^A.S T Ar2.A 2 +... + Aî1jAS1n is equal to Aw,., if n is even, and to - wA s if n is odl. 47. If f (x)= x' + aCt1x-l + x + 2-... -+ = O has for its roots bl, b... b,, prove that 1, bx, b.... b6, b6 b,_l, b,,_l, bXl... X, b, 1, 1, 1... 1, 1 And if s,. is the sum of the rth powers of the roots n?-1.. 1 1 (1*-1) l X l, 2 * =.(-). (b, bS... b6,)f(x). Sn î 8n-. S, 8 8n+1,l î... 8, S1 82i —1 S2 2i-2 ~ 1 ~ ~ t Sn-1 48. Prove that ar a î-2 a 1a a21', =,.-2... I,1)._,+ aC'L a -2... a2 1 a1, 12... a 1 Hp being the sum of the homogeneous powers and products of order p of a1, a2... a,,. 1 1 49. If caris a-, a/s - Cai/. 9-(Xs - y,.)' prove that the value of the determinant of order 2n C11, all... a1l, aUl | a12, a12... aC t2 al2 6t1, 2n, aL1,2 * 2 a,, 2 a^, 121 272 THEORY OF DETERMINANTS [EX. *is. 2(a,, a*. an) 2(x^x.. x2) (- (a) ( (a,)... ) (a,) ]2 where (b (x) = (- x1) (x - x)... (x - x?). 50. Prove that the value of the determinant of order 2n + 1 whose ith row is 1, sin ai, cos a, sin2ai, cos 2ac,... sinnai, cosnai, is 22n9- Il sin 2 (ai - a), where i, k are all duads from 1, 2... n (i > k). Also that the value of the determinant of order 2n whose ith row is sin, cossin 2ai, co s a., sn., 2a.. sin nai, cos nai, is 221-22+-1 I sin 2 (ai - a1) S, where S= S cos ~ (aC + a9 +... + aC, - a,,... - a2,) is formed by dividing the 2n angles into two sets of n in all possible ways and taking the cosine of half the difference of the sums of these sets. 51. If A — 1 i 1_ 1 1........................................... Ca- - Xn+ a2 - xz+l -, prove that -_ ( ~)"t 4' (a', a.-.. *a) ( (x1, x2... * + ) (a) 4 (a,,)...4 (aj) where 4) (x) (x - xi) (x - x2)... (x - i+l) If B is the determinant obtained from A by writing (a,. - )2 in place of (a - x), prove that A I-1 - 1 - i B ai-x 5 a. - x 1. - X2 1 1L-.. 1 I aL1-,X2,t-i, a1-x9 ' ai - 2 a 62 -- 2 ai, -_ l2 a 1 x, 1 ax (\ - 'gq+l - 6- — + 6 -+l C2(+ - 'zt+l /- 1 49-52] EXAMPLES ON THE METHODS OF THE TEXT 273 the function on the right being formed like a determinant, with all the signs positive instead of alternating. 52. If a, 3... A; a', /'... ' are two sets each of n quantities, and Cr is the product of ail the binomial coefficients in the expansion of (1 + x), prove the following equalities: (a- a'), (a- 3')"... (a - X) C | ~~~~(at-a )'r (23- (3-) = ( ^KX) (a, )... )............................... (X-a')n, (A_:')n À(X_ ')n where I= fa-a', a-.3'. a - J a3-', -'... P -x' Ax-a', X-/ 3... A-A' If u = (x - ay) (x - 3y)... (x - Ày), v= (x- a'y) (x - 3'y)... ( - À'y) i= (12)"^v, using the notation of invariants, (a - a')"... (a- À')", (a - )" = (- 1)nC,. ) (a,... ) (a,. X') uv.................................... ( - a')n... (X- ')", (X- x)n (ax-a')"... (a - X')n (,a-, '+ +...(, (-x'p+~, in,+- _ x(+............................................. =(- ( ).(...()f) - GCl)n+l ~~ ~ (X (ÀX ) P (x- a')+... (x - x')+, (x - x)"+' x (a', 3'... À') I. Uv, (x -a'),+'... (x- x)-+~ where I= a-a... a-X', a-x =-(1 2)11-1. P-a'... 3-x', I —x a... X x kx - a... a - ' j Again, i... 1 S. D. 18 274 THEORY OF DETERMINANTS [EX. (a -a')n+l... (a -X')+ i (_ )l Cn~ ) (................................. 4 ) (À - a')+... ( - À')"+, 1 i 1... 1 I 1 9 I'-=a-a'... a-À', 1 - a'.. - X', 1 1 ' 1 Cdu dv (12 dx dx, 53. Let there be two systems of binary n-tics u1... un; v1... v where ui = ao, x + nqaxliX"- y +i- 't2aX2i'- +... + aiy vi = boi x + nXlbi-1-y + n2b21xy1-2Y2 + b,+ ny". And let (i, k) be the lineo-linear invariant of ui and %, so that (i, k) = aoi bk - nlali bn-ik + 22a2ibn-2 -... ~+ anibok Prove that (1,1).. (1, +2) -0, o................................. (n+92, 1)... (n +2, n+ 2) (1n +, 1)... ( 1,+ 1) ao+, l,,. a.. bo.b bln... b 54. If al, a... a, are the roots of the equation Xn + px -l+... + Pn = prove that d(Pl, P... P.) d (a1,, (.. a,,) 55. If u =- x2=-.. -=Xnxn Xn Xn x, being a function of xl,,2... xn- given by X1 + X2 + X_122 +,,2 +, prove that d(ul, U2... un-l) 1 dc (X)I 2... n-i-) l' 52-59] EXAMPLES ON THE METHODS OF THE TEXT 275 56. If u =(x+y+z) + (x-y - -z) (-x + y-) + (-x-y + ), prove that the Hessian of u,, is u2_- (x4 + y4 + ^4 _ 2x~y2 2y2%2_ )-2 multiplied by a numerical factor. 57. If F=1u2u... u,, where ui, u2... u, are linear functions of the n variables x, x2... x, prove that F2f (log y) =-(-1) [d(l z. X), u..)] (x(, x...:' Also that F dF dF _/ ^_id (U..)12 F. d^ '". dCI = (_... X1ll) dF d2F d2F dxI' dx,2 '. dxdxn dF d2F d2F dx,' dx, dx ' ' dx,,2 58. If u1, u2, u3 be three functions of x, y, and if c (U, %3) d (vu, u1) cd (u, u2) vl d(x, y) V 2- d(x, y) -3 d (, y) d (vy, vy) w = V2, 3&c., d(x,y) prove that W1 W2 W3 u'1 u2 U3 59. If ul, u2, u;, U4 are four functions of x, y, and if v = d2u, d2u3 d2u4 dx2 ' dx-2 dx2 d2,u2 dCU d 2a4 dxdy dxd ddy ddy d2u2 d2u3 d2U4 dy2 ' dy2, dy2 and V2, 3, v4 similar determinants formed from u., u4, Uz, &c., then 18-2 276 THEORY OF DETERMINANTS [EX. from v1, '2,, 4 we can form four new functions wl, w2, w3, w4 in the same way as we obtained vl... V4 from t1... 4. Prove that Wi d3lu d3u2 1d3u3 d3u4 U~ dx~ ' dx ' dx3 dx' d13ul d3u2 d3u1 d3u4 dx2dy ' dx2dy' dx2dy' dz dy d3U1 d3u2 d3u3 d3u4 dxdy2' dxdy2 dxdy2' dxdy2 dIu1 d1u2 d.u3 d( 'u4 dy3 ' d ' dy3 dy3 where u is a numerical factor. 60. For the n2 functions ik (i, k 1, 2... n) of the variables x1, x2... xî, prove that the cubic determinant whose elements are du\ dd ik (z(,,l, 2... r ) is a covariant. 61. For the n functions u1... u, of the variables xi... xn, prove that the cubic determinant whose elements are d21i dxjdxk (,' j, k= 1, 2... n) is a covariant. 62. If the function u of the variables xi... xq be transformed by the linear substitution xi = biyl + bi2y2 +... + bin-iYto a function v of n - 1 variables, prove that H (v)=- 0, Bi... Bn -B1, Un... ~iln Bn 5 Uîil... Unn d2u where u k —,d, and (- 1)iB is the determinant obtained by suppressing the ith row in the array formed by the quan ities bi. 63. If u = aaikXiXk (i, k =1, 2... and Dr = all,... air ar1... arr 59-66] EXAMPLES ON THE METHODS OF THE TEXT 277 prove that the substitution 1 dD,.+ 1 dDn xr~ = ~r + +1 _ n - û rw =y+ Dda ++ D- daY+ +) reduces the given quadric to the sum of the n squares Dr64. If u and v are two n-ary quadrics and U, V their reciprocals, prove that we can by the same linear substitution change u into A V and v into BU; A and B are the discriminants of u and v. The determinant C of the substitution is the geometric mean between the discriminants of U and V. If C be regarded as the discriminant of a quadric W, we can by the same linear substitution reduce the three quadrics U V,,W to the sum of squares. The coefficient of any term in W so transformed is the geometric mean between the homologous coefficients in U and V. 65. If to the leading elements of the determinant of an orthogonal substitution of order n we add the quantities a1, a2... a., or the quantities,...-, the resulting determinants are equal if a1 a. a. ai a2... a, 1. 66. If ci. are the coefficients of an orthogonal substitution (modulus unity) of order n, prove that D= cll-., c12... C1 C21, C22-1 i. C2%............................. CnI, Cn2... nn - is equal to zero if n is odd; but if n is even its value is 27& [A] A ' where A is the skew determinant from which the orthogonal substitution is derived, and [A] the same determinant with the elements in the leading diagonal zero. If D~ is the coefficient of one of the leading terms in D, prove that when n is even 2Di - D. 278 THEORY OF DETERMINANTS [EX. 67. If cik l=E is the determinant of an orthogonal substitution, the equation Cll + X, Cl2... C1, = C2, C C22 +-... C2n.............................. Cnl 7 cn2 ' ' Cnn + X is a reciprocal one. If n is odd it has one real root- e; if n is even and c =- 1 it has the two real roots + 1. The rest are all imaginary. 68. The maxima and minima values of Uc =:CikXiXke subject to the conditions -= bikXiXk C111 + C12x2 + *.. + ClnX2 = O Cn-21X1 + Cn-22X2 +... + 0Cn-2ngCX = O are given by the equation bllu - allv... bhn u-an v, cil, c2,... c_2 =0..................................................... b.l u - alv... bnnu - ann ), CL, C2n.. Cn-g2b C11 '.. Cin...................... Cn-t21 ~.' Cn-2n 69. The values of x1, x... x, which satisfy the equations a11xl + a21 X +...+ anx m = O a1_-l, + ct2,_lX2 +... + 6mr,-lx - = O Calx1. + a2,.x2 +... + amîx [ = 1 alr+l X1 + Ca2r+l X2 + ' -+ anmr+l Xm =.......................................... al,,x1 + aC2... + Camî+ mxm = and make x x2 + x +... + x2 a minimum are 1 dC 1 dC 1 dC 2Cdalr ' 2C da ' 2C dar' where C is the deterin inant whose elements are given by Cik= alialk + a2ia2k +..-.- a1iamk 67-72] EXAMPLES ON THE METHODS OF THE TEXT 279 70. The value of the integral... xi x dxdx2... dxn taken for all values of the variables such that axijxiXj = < 1, the quadric being a definite positive form (i.e. incapable of becoming negative), is r (+ 2) 2A where A = [ ak \ is the discriminant of the quadric. 71. The value of the integral f... Y eE-cos (b x1 + b22 t... + bxn) dx dx2... dx, where î6 = -a-ik Xi Xk) i& E 4E where 1 O, bi, b2... bn A bl, all, a.12 a,, bn, a1,l a... aCn In this question and the next u is supposed to be incapable of becoming negative. 72. The value of the integral r~oo oo v... v ve-dx1 dx2... dx., o -o oo where v -b= SXik?1, = a ikz iaXk, //7rnS is is 2/ 4A3 22' where S is the sum of the n determinants obtained by substituting for each column of A in succession the corresponding column of the discriminant of v. 280 THEORY OF DETERMINANTS [EX. 73. Let a%, a2... a2,+1 be 2n + 1 real and different numbers in ascending order of magnitude, and let P (x) (x - ia) (x - a.)... (x - a,,+1) Q (x) =(x- c2) (x - a4)... (x - a,,) A R (x)= P () Q (x), A being a positive number. Then if K [a2s P (x) dx L Q_ Q (_r-) fa.s P (x) dx J 2J a2 (x - a2 (X) R(x)' - 2 p (a2r-1) J a21 (x - 21)2/ R( () (these are the complete Abelian integrals of the first and second species), and if also _/1 ra2s8+l P (x) dx!-i Q (2l) [)2s+ P (x) dx 2 J a2 (x- a62t1)R(Z)' IS 2 P' (a21.-)J as (x-asrl)2\/R(x) then v t t v lu o t cntinu in D = kU, 11 * * * /?l =1 (2 \, Kn, L11 ' K'nn 5 ad L in f kln ~ li... kjjj,, ï7t1 Prove also that dD _dD 7 n6-l dD dD _a r cdK.s,-f, s-l \2/ s-a' s dL?" s-l \ kr sdD dD __ iL-L dD dD i j dkr, sl dksl ~ 2 d s-l, - d ~ ) r S- 1 74. Prove that the value of the continue d fraction a b c --- ad a llf. a + 1 - i + 1 --- -- is unity. 75. Prove that the product of the two continued fractions 12 32 - + )(a -2+ -2 (a- 1)' + 1P 32 2 (a+ 1)2+ 2 (a++ 1) + is a2. 73-80] EXAMPLES ON THE METHODS OF THE TEXT 281 76. If un is the number of terms in a determinant of order n which do not contain any element from the principal diagonal, prove that n = _i + (- 1)+, and hence that n is the coefficient of xn in the expansion of -- 77. If un is the number of terms in a symmetrical determinant of order n, prove that 1(n- l) ( - 2) n - nu^ - -^- 2 u u 0. Also that -n is the coefficient of xn in the expansion of n! elx+ x2 J(1-x)' 78. If [1.3. 5... (2n- 1)] u is the number of terms in a skew determinant of order 2n, prove that u, -- (2n - 1) zun_ - (n - 1) U_2. Shew also that 2 -- is the coefficient of xn in the expansion of 79. If A is the area of a quadrilateral, the co-ordinates of whose angular points are (x1, y,)... (x4, y4), then 1, 0, x, y, 1, O,,, - X4, -X, Y4-Y 1 0, 1, X4, y4 The area of a quadrilateral inscribed in a circle in terms of its sides is given by 16A=- -a, b, c, d b,-a, d, c c, d,-a, b I d, c, b,-a 80. If the planes aix + biy + cz + di, = (i = 1, 2, 3, 4, 5) touch the same sphere, then ai, bi, ci, di, u - (i= 1, 2... 5), where U2i= ai2 + bi2 + Ci2 282 THEORY OF DETERMINANTS [EX. 81. A quadric of revolution passes through five points P1... P, and the distances of these points from a focus are r... r5. If V1 = volume of tetrahedron P2 PP, &c., prove that Vr + 2r2 +... + V5r= 0. 82. Let V, V' be the volumes, A, B, C, D; a, b, c, d the areas of the faces of two tetrahedra whose angular points are numbered 1, 2, 3, 4. Also let Pik be the perpendicular from the point i of the first tetrahedron on the face opposite the point k of the second, and pk a like quantity for the other tetrahedron. Prove that Pik 1 x ik i =: ACDabcd (i k= 1, 2, 3,4). 83. If A, B, C, D are the directions of four forces in equilibrium, and if AB is the moment of the lines A and B, &c., prove that O, BA, CA, I)A =0. AB, 0, CB, DB AC, BC, 0, DC AD, BD, CD, O If a, b, c, d are the magnitudes of the forces a= I(BC. CD. DB), &c. 84. In Siebeck's determinant, xvII. 22, prove that dD = 288vv, ddik where v is the volume of the tetrahedron formed by the face opposite the point i of the first tetrahedron and the centre of the sphere circumscribing the second tetrahedron, and similarly for v'. 85. If in a system of five points dik is the square of the line joining the ith and kth points, and r is a sixth point of the system, prove that d,2, dld.2 + d12... drd,,5 + d5, dcl +1 = 0. dridr2 + dl2, dr22... d.2dr5 + d25, dr2 + 1 dridr5 + d15, dr2dr5 + d25... d52, dr + 1 dr, + l d.2 + 1... c+, 1 81-88] EXAMPLES ON THE METHODS OF THE TEXT 283 86. If in a system of seven straight lines, nik is the moment of the ith and kth lines, and r is an eighth line, prove that Mrl2, m1M2 -+ 9%1l2 + nl... 1mr7 + m7 0. mrl 7nr2 + 1 l12, r22.. mr2 1r +-7 -+ 227...................................................... mrl] îr7 + m17 mr2 + l7 +/27 m2 ' 87. Having given two tetrahedra whose angular points are marked 1, 2, 3, 4, let dik denote the square of the distance between the ith point of the first and kth point of the second tetrahedron. Prove the following relations: (i) For two points P, Q the distances of P from the angular points of the first tetrahedron being ai, of Q from those of the second bi, and d = PQ2 d, 1, b1... b4 O. 1, O, 1... 1 ~a1, 1, dl... dl4....................... a4, 1, d41 *... 44 (ii) For the point P and a plane, qi being the distances of the vertices of the second tetrahedron from the plane, p the distance of P from the plane, p, 0, ql ' q4 =0. 1, 0, 1...1 al, 1, dll... d04....................... a4, 1, d41... d44 (iii) For two planes, pi, q, being the perpendiculars from the angular points of the tetrahedra on them, c the angle between the planes, — cos, O, q1... q4 =0. O, O, 1... 1 Pi, 1 d11... l14 p4, 1, C41... d44 88. For a systeni of six and a second system of five spheres, if pi is the power of the ith and kth spheres, 1, Pln.. Pl5 =0. 1, P61... P65 284 THEORY OF DETERMINANTS [EX. 89. The equation 0, S,, 2 S,, 4 = 0 S1, 0, t12, t13S t14 S4, t41, t42, t43, 0 represents two spheres touching the given spheres S =0... S4=0; ti is the square of the common tangent to the ith and kth spheres. 90. Prove that for any five spheres S = 0... S = 0, O, 1, 1...1, 1 =0. 1, 0, t2... t15, S1, t21, 0... t25, 2 1, t5l, t52... 0, S0 1, sl, S'...- S, O 91. The index of two points being defined as in xvII. 27, the index of two planes D, D' is obtained by taking in the planes the points abc, a'b'c' and forming the determinant 1 Iaa,, ' ab', Iac' Dy- 4abc. a'b'c' Iba' ',, i -, L, Ica', Icb' Ice' and the index of two lines y, y' by taking in the lines two points ab, a'b' and forming the determinant I ~ 1 ~__1_ I'aa', 1 ab' '' ab. a'b' a,', Ibb, Prove that for two groups of planes numbered 1... 5 In... I =0,........... l51... I5. 1 (3 V)3 (3V')3 11 '.. 14 A ( ' AB'C'D" (abc)' 2ABCD 2A' B'C'D " 41... 44 where a, b, c are now the semiaxes of the ellipsoid, V, V' the volumes, and A..., A'... the areas of the faces of the tetrahedra formed by the planes. 89-92] EXAMPLES ON THE METHODS OF THE TEXT 285 Prove also that for two groups of lines passing through the points P, P' Il,... I O, 41... 44 I.. 3 - sin (123) sin 1'2'3') Ili. 113 - - - - P?'. (abc)2 131 *.. 13 92. If between the points of two surfaces we establish the correspondence - (x, y, ), -3N = (x,, ), -=x (x, y, z), prove that the ratio of corresponding elements of the surfaces is given by doc de de de ds dx' dyg dz' dr] dr) dr1 d dt cl dx' dy' d d' d a, b, c where (a, b, c), (a, 3, y) are the direction cosines of the normals to ds and dro respectively. HISTORICAL NOTE. THE germ of the theory of determinants is to be found in the following passage, of unknown date, which occurs in one of the manuscripts of Leibnitz left unpublished at the time of his death. "Inveni Canonem pro tollendis incognitis quotcunque sequationes non nisi simplici gradu ingredientibus, ponendo oequationum numerum excedere unitate numerum incognitarum. Id ita habet. Fiant omnes combinationes possibiles literaruni coefficientiunl, ita ut nunquam concurrant plures coefficientes ejusdem incognitoe et ejusdem sequationis. Hæe combinationes affected signis, ut mox sequetur, componantur simul, compositumque oequatum nihilo dabit sequationem omnibus incognitis carentem. Lex signorum hsec est. Uni ex combinationibus assignetur signum pro arbitrio, et cseteræe combinationes que ab hac differunt coefficientibus duabus, quatuor, sex, etc., habebunt signum oppositum ipsius signo; que vero ab hac differunt coefficientibus tribus, quinque, septem, etc., habebunt signum idem cum ipsius signo. Ex. gr. sit 10 + llx + 12y =0, 20 + 21x + 22y =0, 30 + 31x + 32y = 0, fiet 10. 21. 32 - 10. 22. 31 - 11. 20. 32 + 11.22.30 + 12.20.31 - 12.21. 30 = 0. Coefficientibus eas literas computo, que sunt nullius incognitarum, ut 10, 20, 30." (Leibnizens mathenmatische Schriften, ed. Gerhardt, 2 Abth. iii. pp. 5, 6.) Leibnitz then gives a method for finding the resultant of two equations in one variable which is identical with that usually known as Euler's (Salmon's HRgher Algebra, Art. 81). In a letter to De L'Hospital dated 28th April 1693 (l.c. 1 Abth. ii. pp. 229, 238-40, 245) Leibnitz gives, in a modified form, his rule for finding the resultant of a system of linear equations, after explaining HISTORICAL NOTE 287 his notation for the coefficients. It is clear that Leibnitz had realised the advantage of a double-suffix notation for an array of coefficients, and that he had discovered a general rule for constructing what we should now call the determinant of the array; but his rule for determining the sign of each term is expressed in an obscure and clumsy way, both in the passage above quoted, and in the letter to De L'Hospital. This letter, apparently, was not published until 1850. After a long interval, the subject made a new start, initiated by Cramer, Bézout, and Vandermonde. Cramer gave a rule for writing down the solution of a general set of linear equations; Bézout's principal contribution is the relation | c,,, I= -asAis and its immediate corollaries. Vandermonde is really the founder of the calculus of determinants, in the proper sense of the term. He gives an independent definition of a determinant, denotes it by an appropriate symbol, connects the rule of sign with the permutations of either of two sets of indices, and obtains the expression of a determinant of order 2m as the sum of products of minors of order 2 and minors of order 2m, - 2. Thus Vandermonde partly anticipates the theorem, usually known as Laplace's, on the expansion of a determinant in terms of conjugate minors. The actual term determinant (in its Latin form determinaozs) is derived from'Gauss's use of it to denote the discriminant of a quadratic form. It was introduced, in its more extended sense, by Cauchy, whose memoir in Journ. de l'Ec. Polyt., x. cah. 17, pp. 29-112 (1812) contains, except in the matter of notation, proofs of all the most fundamental formal properties of determinants. This and the memoir of Binet (ibid. ix. cah. 16, pp. 280-302) exhibit the theory in a developed form, and complete the first stage of its history. In 1825 F. Schweins published a work (Theorie der Differenzen und Differenziale, u.s.w. Heidelberg) which contains very important results relating to compound determinants. Unfortunately his notation is very cumbrous: his work received little or no attention, and his results were rediscovered by Sylvester, Kronecker and others. The discussion of the arithmetical properties of determinants and the theory of elementary divisors is of comparatively recent origin: the principal contributors to the subject are H. J. S. Smith, Weierstrass, Kronecker, and Frobenius. In 1877 G. W. Hill published an important memoir on lunar theory in which he made systematic use of determinants of infinite order (reprinted, with additions, in Acta Math. viii. pp. 1-36). The 288 HISTORICAL NOTE properties of such determinants have been discussed by H. Poincaré, Helga v. Koch, and T. Cazzaniga. T. Muir's Theory of Determ7inants in the Historical Order of its Development, Part I. (London, 1890), gives a most careful and complete account of the progress of the theory down to the year 1841. Dr Muir has also compiled a list of writings on determinants, two parts of which have been published in the Quart. Journ. of Math. vols. xviii., xxi.; the third part, coming down to 1900, will shortly appear. CAMBRIDGE: PRINTED BY J. AND C. F. CLAY, AT THE UNIVERSITY PRESS.