26 PEOFESSOE CATLEY ON THE THEOBT OF MATEICES. 



Article Nos. 3 to 13. — First Investigation. 



3. We have to consider the formulae for the automorphic linear transformation of the 

 function ww'-\-yz'—zy' — wx', that is, of the function 



( • , • . • , —^ Xx, y, z, xoXp(/, y, z', -m/) 



• J • ■> ^■i 



• ) -'•> '5 



1, . , . , 



={Q.Jjc, y, z, wXx\ y, z', w'), 

 viz., if the variables are transformed by the formulae 



(;r,3/,z,w)=(nXX,Y,Z,W), 



(:r',y,z',i<;') = (nXX',Y', Z',W'), 



then the matrix (11) is such that we have identically 



(QXo:, y, z, w^x, y', z\ w')=(QIX, Y, Z, WJX'. Y', Z', W) ; 



the expression for (11) isgiven in my memoir above referred to ; viz., observing that the 

 matrix (Q) is skew symmetrical, then (No. 13) we have 



n=Q-(Q-Y)(n+Y)-'Q, 



where T is an arbitrary symmetrical matrix. 



4. I propose to compare with the matrix 11 the inverse matrix 11"'. Recollecting 

 that in the theory of matrices (ABCD)-'=D-'C"'B-'A-', we have 



n-' = £2-(Q+Y)(0-Y)-'Q; 



and it is to be shown that 11 and 11~' are composed of terms which (except as to their 

 signs) are the same in each, so that either of these matrices is derivable from the other 

 by a peculiar form of transposition. It is to be borne in mind throughout that T is 

 symmetrical, Q skew symmetrical. 



5. I write for greater convenience 



-n =Q-'(T-Q)(T+0)-Q, 

 -n- = Q-'(T+Q)(T-Q)-'Q, 



and I compare in the first instance the matrices (T— 0)(T+n)~' and (T+n)(T— 0)"'. 



6. Any matrix whatever, and therefore the matrix (T + Q)~', may be exhibited as the 

 sum of a symmetrical matrix and a skew symmetrical matrix ; that is, we may write 



(T+Q)-=T'+Q', 



where T' is symmetrical, Q! is skew symmetrical. AVe have then 



(T+Q)(T+Q)-'=(T+0)(r+Q')=l, 



