PROFESSOE CATLEY 0>s^ THE THEOEY OF MATEICES. 

 where, here and in what follows, 1 denotes the matrix unity. Moreover 



T-Q=tr.(T-fQ), 



(T-Q)-' = (tr.(r+Q))"' = tr.(T+n)-' = tr.(T'+a')=Tr'-^'; 

 (T-Q)- = T'-Q'; 

 (T-fi)(T-Q)-' = (r-Q)(T'-0') = l. . 



27 



and thence 



that is, 



and thence also 



We have therefore 



(T-Q)(r+Q)-'=(T+Q-2QXT^'+^')=l-2ii(T'+Q'X 

 (T+QXT-Q)-'=(T-n+2Q)(T'-Q')=l+2Q(T'-Q'). 



7. Suppose for a moment that 



r' + Q'=( a, b, c, d) 



i , y, k, I 



m, n, 0, p 



T'— Q'=( a, e, i, m ) 



f>, f, J, n 



c, ff, k, 



d, h, I, p 



and therefore 



8. We have 



-n(T'+n')=( 



. -1 



-1 



1 )( a, b, c, d ) 



e , /, g. h 



i , J, k, I 



m, n, 0, p 



. -1 



-1 



i> )• 



I 



m, n, 0, 



i , y, A-, 



— « » — /. -5^' --^* 

 — a, —b, -— c*, — (Z 

 E 2 



