TABER. — TRANSFORMATION OF A REAL BILINEAR FORM. 319 



Conversely, every transformation of tlie first kind can be generated by 

 an infinitesimal transformation of V'". For let T be any transforma- 

 tion of r"', then 



(10) f=ATA~\ 



Let F"and W V — 1? respectively, be the real and imaginary parts of the 

 polynomial U =f(T) satisfying the equation T = e u . Then, since T 

 is real, both V and W are polynomials in T, and therefore commutative. 

 Let 



T x = e v , T = e wV=r \ 



Then since V, and therefore T x = e r , is real, and since 



T= e T+wV=i = e v e wv '~ l = 7\ T 



is also real, it follows that T = T x ~ x T is real ; and therefore, by 



Theorem I, 



T 2 = (e wV ~ 1 ) 2 = 1. 

 Wherefore, 



T 2 = (T 1 T ) 2 = 71 2 T 2 = T,\ 



But, since V =■ 4> (T) is a polynomial in T, we have by (10) 



V = <£ (f) = $ (A TA- 1 ) = A<f>(T) A~ l = A V A~ x ; 



and therefore, since J 7 is real, e^ r for any real scalar £ is generated by 

 the infinitesimal transformation e ^ r of Y'" . In particular, 



. T 2 = T x 2 = {e 7 ) * = e 2v 

 is generated by e^ 7 . 



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

 equation of T has no negative root. For then, by Theorem II, there 

 is a real polynomial U = f (T) satisfying the equation T = e D . In 

 this case V = U, W = ; and U satisfies equation (5). Therefore T 

 is the second power of the transformation ei^of r'", and is generated by 

 the infinitesimal transformation e $ u of T'". 



The characteristic equation of a transformation of the second kind 

 may have negative roots. But in such case the numbers belonging to 

 the negative root of this equation are all even.* 



Every transformation of r'" with negative determinant is a transfor- 

 mation of the second kind. Let n = 2, and let 



* Cf. p. 312. 



