280
ALGEBRAIC
and from these another equivalent set A&aq = (2Bn — + 2B12X2 + 2B13X3 + ..., Aja;2= 2B21X1 + (2B22 - A6)X2 + 2B2JJX3 + ..., and now writing
FORMS
tion ; but if A vanishes the equations can be satisfied by a system of values other than zeros. For in this case the n equations are not independent since identically Alfi/l + A 2^/2 + ... + Annfn = 0, and assuming that the minors do not all vanish the satisfaction of % - 1 of the equations implies the satisfaction of the nth. Consider then the system of 71 - 1 equations a 2yXy + *22*2 + • • • + *2n*n = 0
- 31*1 + *32*2 + • ■ • + *3n*n = 0
2Bii-Aj 2Bi4 T = ««, .— = ««, we have a transformation which is orthogonal, because SX2=Sa32 and the elements an, aik are functions of the {n - 1) independent quantities b. We may therefore form an orthogonal transformation in association with every skew determinant which has its
- ?U*'i "b * ?i2*2 ~b • * • _f_ CXnnXn — ,
leading diagonal elements unity, for the (n- 1) quantities b are which becomes on writing ^ = ys, clearly arbitrary. For the second order we may take «2l2/l + *222/2 + • • • + *2!«-l2/»-l + *2n = 0 4
- 3l2/l + *322/2 + • • • + *3)n-l2/n-l + *3» = 0
- = | -m| =1+x!and the adjoint determinant is the same ; hence
- nl2/l + *n22/2 +•••+*«, n-l2/n-l + ann = 0.
(l+A22)aq = (l-X2)X1+ 2X , We can solve these, assuming them independent, for the n-1 2 (1 + X )a;2 = - 2XXx + (1 - X2)X2. ratios yx, y2, ...y„-y. Now Similarly, for the order 3, we take
- 2lAii
+ *22A12+ ... +«2nAi„ = 0 1 V-fl a 2 2 2 31-^-ll +
- 32 A12 + • • • + *3n Ai„ = 0
A6 = -v 1 = l+X + /i + »/ H-X 1
- «i
A + a u n2Aj2 +... + a ,in A Iri — 0. and the adjoint is and therefore, by comparison with the given equations, Xi=pAyit 1 + X2 v + fi2 — fi + v where p is an arbitrary factor which remains constant as i varies. - v + X^t 1 + ^t
+ [XV
[x + v - +[XV 1 + p2 Hence y^— where Ayy and Ain are minors of the complete leading to the orthogonal substitution 2 2 2 determinant (*11*22-• •*»»»)• AbX1 = (1 + X - /a — »' )X1 +2(v + Xm)X2 + 2( — /r + Xj')X3 '22 1 *2,i+l ...*2n AbXy — 2(/x — v)X1 + (1 -h fx2 — — v2)X2 +2 2(/rv + X)X 3 '32 ••-*3)1-1 *3)i+l ■••aSn Ai,x3— 2(v + /x)X1 + 2(yU.v — X)X2+(1 + v — X2 — ya2)X3. Functional determinants were first investigated by Jacobi in a . . / M '.-n
- «2 -»-*«) 1—1 *n)i-f-l ...Unn
work De Determinantibus Fundionalibus. Suppose n dependent • • Vi ) .. •• -*2)n—1 _ ) n 1 *22 variables y-^, y2,each of which is a function of n independent L *32 •••*3)n-1 variables aq, x2,...xn, so that x2,...xn). From the differential coefficients of the y’s with regard to the afs we form the
- nl *n2 • • • *n) n—1
functional determinant or, in words, yyth is the quotient of the determinant obtained by ojh 9Fi erasing the i column byi+nthat obtained by erasing the nth cx2 dxn column, multiplied by ( - l) . For further information concern9^2 9^2 = fHv y*■■■&») ing the compatibility and independence of a system of linear R = dxx cx2 dxn {x^ x2,...xn] equations, see Gordan, Vorlesungen iiber Invariantentheorie, Bd. for brevity. 1, §8. . CVn 9?/n ^Vn Resultants.—When we are given k homogeneous equations in k (V'] ex.. dxn variables or k non-homogeneous equations in A;-1 variables, the If we have new variables z such that Zs = <&(2/i> Vif-Vn), we equations being independent, it is always possible to derive from have also z3 = j/s(x1, x2,...xn), and we may consider the three de- them a single equation R = 0, where in R the variables do not terminants appear. R is a function of the coefficients which is called the “resultant” or “eliminant” of the k equations, and the process fVlt y2>--yn fzlt z2,---zn' fZy, z2>-,,2n' by which it is obtained is termed “elimination.” We cannot
- 1. X2,,..XnJ /i, y2,---ynj V^i, 2!2, .^.. Xn J
Forming the product of the first two by the product theorem, combine the equations so as to eliminate the variables unless on the supposition that the equations are simultaneous, i.e., each of we obtain for the element in the ith row and kth column them satisfied by a common system of values ; hence the equation dzi "dyi ^ c~i 9% 9y« R=0 is derived on this supposition, and the vanishing of R exdyy dxk cy2 cxk + ... + 5-7 UVn dxk’ presses the condition that the equations can be satisfied by a which is g^-, the partial differential coefficient of zt with regard common system of values assigned to the variables. Consider two binary equations of orders m and n respectively to xk. Hence the product theorem expressed in non-homogeneous form, viz. /(*)=/=a^x™n - a^x™-^ + a2xm~2 - ... = 0, (zli yy, y^) •••yn— ( z, 22,... 2n . ip[x) = <p = b0x - 2qa5n + b.jX™ ~2 - ... = 0.
2/l) y^y • • • 2/w/ *^1) X2,... Xn J *^ii *^21 •• • •*'«/
and as a particular case If a1,a2,...am be the roots of/=0, /32,the roots of 0 = 0, the condition that some root of 0 = 0 may cause / to vanish is 2/i, 2/2,--^nVXy, a?2,...a’n,_1< clearly x2,...xnJ y3, y2,...yn) Kr,<t>=APiW2)-f(Pn) = 0-; Theorem.—It the functions yx, y2,...yn be not independent of so that R^,^, is the resultant of /and 0, and expressed as a function one another the functional determinant vanishes, and conversely if the determinant vanishes, y3, y2,...yna,Tv not independent func- of the roots, it is of degree m in each root /3, and of degree n in each root a, and also a symmetric function alike of the roots a and tions of aq, a?2, Linear Equations.—It is of importance to study the application of the roots /3 ; hence, expressed in terms of the coefficients, it is of the theory of determinants to the solution of a system of linear homogeneous and of degree n in the coefficients of /, and homogeneous and of degree m in the coefficients of 0. equations. Suppose given the n equations Ex. gr. fi:= — QyyXy f «12a;a;2 + ... + aynxn—0, f= agx2 - ayx + a2 = 0, 0 = b0x2 - b3x -f b2. /b ot2yXx + a22 2 + • • • + a2nXn — 0, We have to multiply a0/3j -a^ + a., by a0(3?, - axp2 + a2 and we obtain fn — 0'nXy + Un2X2 + ... + annXn = 0. a2/32£2 - a0*i(£iA3 + /3i/3|)+ *0*2(/3i+/3|) Denote by A the determinant (a11a22...ann). Multiplying the equations by the minors Aifi,A2/Ji,...A„^ re*i*2(/3i + ft) + a I > where spectively, and adding, we obtain hl X^aypAyfi + a2fnA2iJi + ... +anijL Ann) = XlxA = 0, ft+ft=p bf° °0 ftft=p °0 Pl+Pl = ~/'o since from results already given the remaining coefficients of and clearing of fractions x1, x2, ...Xpt-i, X/x+y, ...xn vanish identically. Hence if A does not vanish ^ = *2=... =ajn = 0 is the only soluR/,<#>=: (*0^2 — *2^0)* (*1^0 — *0^l)(®1^2 ~ *2^l)-
