, 



— m, —i, e, a 



34 PEOFESSOE CATLEY ON THE THEOET OF MATEICES. 



=( . . . -1XX,Y,Z,WXX',Y',Z',W') 



. . -1 



. 1 

 1 



=XW'+YZ'-ZY'-WX'; 



and similarly writing 



(X,Y, Z,W)=( p, l,-h,-dXx,y,z,w),{SJ,Y',7J,W)=.{ p 

 , k, —g, —c 



—m, —i, e, a 



we obtain with the s coefficients the equivalent result, 



XW'+YZ'-ZY'-WX'=arw'+2/s'-zy-wa:'. 



We thus see conversely that the Hermitian matrix is in fact the matrix for the auto- 

 morphic transformation of the function xv^ -\-yz' — zy'—ws^. 



22. Considering any two or more matrices for the automorphic transformation of such 

 a function, the matrix compounded of these is a matrix for the automorphic transforma- 

 tion of the function — or, theorem, the matrix compounded of two or more Hermitian 

 matrices is itself Hermitian. 



Article No. 23. — Theorem on a Form of Matrices. 



23. I take the opportunity of mentioning a theorem relating to the matrices which 

 present themselves in the arithmetical theory of the composition of quadratic forms. 



Writing 



l,—1i, —dx^,y\ ^, W), 

 k, —g, —c 



(X)=( 



a , a , 5+/3 ) 



— a , . , 5-/3, c 



and.-.(X)-=^( 



and .-. (X')-=5^/ 



-a ,-{b-^), . , y 

 -(*+^), -c , -y, . 



where D=cc— i^ A=ay— /3^; and similarly, 



X'=( . , «',«', 5'+i3' ) 



-«' , . , *'-j3', d 



-a! ,-(b'-(3'), . , y> 



v/hevel>'=a'(/-b'\ A'=a.'y'—I3'^; then 



(XXX')+(D-A)(D'-A')(X'-XX-'), 



. , y , —c , b—ft ) 



— y , . , b+ft, —a 



c ,-(*+i3), . , a 



•(5-/3), a , — a, . 



. , y> , -c' , b'-(i' ) 



-y' , . , b'+fi', -a' 



.(i'-/3'), a! , -u, . 



