AUTOMORPHIC LINEAR TRANSFORMATION OF A QUADRIC. 



213 



where D is the matrix corresponding to the discriminant of the quadric. The latter 

 result is evidently much the more complicated, but the difference between the two is 

 far greater than the outward appearance of the expressions 



(A)- 1 (A') and (D)- 1 (A')(A)" 1 (D) 



might at first suggest. Even in the case of only two variables — the transformation of 

 ax 2 + 2hxy + by 2 — the repeated multiplications implied by (D)" 1 (A / )(A) _1 (D) involve a 

 surprising amount of labour. 



It was soon ascertained that considerable simplification of Cayley's coefficients was 

 possible, and at last it became apparent that in every case they could be reduced to the 

 comparatively simple forms given by the other formula. The consequence, of course, is 

 that when D and A have the forms 



) 



(« 



h 



9 ... 



) and ( 



a 



h+l n 



9 + hz • • • 



h 



b 



/ ••• 





h + l 21 



b 



f+hs ••• 



9 



f 



c ... 





9+ hi 



f+h2 



c . . . 



we can assert the curious theorem in matrix multiplication that 



(D)-XA')(Ay\I)) = (A^XA'). 



(6.) On account of the lengthy expressions involved, the proof is not by any means 

 easily set down if the calculus of matrices be not utilised.* Confining ourselves to 

 matrices of the 3rd order and writing, v , — n , X for l 12 , Z 13 , l 23 , the identity to be 

 established is 



( a h g )~\ a h-u g + /j. )( a h + v g-fx )~\ a h g ) 



h b f 

 9 f c 



h + v b f-X 

 g-/j. f+\ c 



h-v b /+\ 



7+M /-X c 



h b f 

 9 f G 



= ( a h + p g-fj. ) \ a h-v g + m ) 



h-v b f+\ 

 g+H f-X c 



h+v b f-X 

 g-H f+X c 



Now if we denote the complementary minors of a , h , . . . in D by A , - H , . . . , and 

 the complementary minors of a, h + v, . . . in A by cof (a) , — cof (h + v) , ... we know 

 from Cayley that 



* With the aid of this calculus, however, the proof is very simple, and will be seen to hinge entirely on the fact 

 that a+a'=2D . At its fullest extent it stands as follows :— 



D-iA'A-iD = D-i(2D-A)A- 1 D , 

 = (2-D- 1 A)A- 1 D, 

 = 2A~ 1 D-1 , 

 = A" 1 (2D-A), 

 = A-!A'. 



Here A' is the matrix got from A by changing rows into columns : but this relationship is not a necessity for the 

 existence of the identity, which will hold if a , D , a' be any three matrices whatever fulfilling the condition 

 a+a'=2D. 



