TABER. — TRANSFORMATION OF A REAL BILINEAR FORM. 315 



T =AT~ 1 A-\ 



(8) 



T~ l = A TA- 1 ; 



and therefore each root, other than ± 1, of the characteristic equation of 

 Tis paired with its inverse, so that if £, zjz zk. 1 is a root of this equation, 

 &, = 4 -1 is also a root, and the numbers belonging to £. and &, (in par- 

 ticular the multiplicity of these roots) are the same. Therefore, by 

 Theorem III, there is a polynomial f(T) satisfying the equation 

 T=e AT) and such that f(T) -f(T^) is real. Let now 



2 U Q =f(T) +/(7 T - 1 ), 2 U x =f(T) -f(T- x )i 



and let 



T = e% T x = e u K 



We have by (8) 



rpi __ g/lrj+zir" 1 ) 



— e fiT) e fiT' x ) 



— e M) qMTX 1 ) 

 _ g/tT) gAADA" 1 



= 7M r • ^- x = A A' 1 = 1. 

 Therefore, 



t 2 = (t t\) 2 = r 2 r x 2 = t x \ 



But, by equation (8), 



= i(f(AT-iA-*)-f(ATA-i)) 



= $A(f( T-i) -/( T)) A-* = -A U x A-' ; 



and therefore, since U x is real, it follows that e$ l , for any real scalar £, 

 is generated by the infinitesimal transformation e s £ Ui of T". In partic- 

 ular 



T 2 = T l 2 = (e v if = e- l \ 



is generated by e 6 ^ 1 . 



A transformation T of T" is of the first kind if the characteristic 



