TABER. — TRANSFORMATION OF A REAL BILINEAR FORM. 319 



Conversely, every transformation of the first kind can he generated hy 

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

 tion of r'", then 



(10) T=ATA-\ 



Let Fand W V— 1» respectively, be the real and imaginary parts of the 

 polynomial U = f{T) satisfying the equation T = e^. Then, since 7' 

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

 Let 



7\ = e\ To = e'^'^'^K 



Then since V, and therefore T^ = e^, is real, and since 



^_ gT+WV'-l 



^V^WS~l^ rj^^ rp^ 



is also real, it follows that T'q = 71 ^ T^ is real ; and therefore, by 

 Theorem I, 



71,2= (e'^*~)2 = 1. 

 Wherefore, 



T^ = {T^ Tof = 71^ To" = 712. 



But, since V=: cji (T) is a polynomial in 71 we have by (10) 



V=cf>{T) = <j>(A TA-^) =A<i>(T)A-^ = A VA-' ; 



and therefore, since Fis real, e^^ for any real scalar ^ is generated by 

 the infinitesimal transformation e^^'' of T"\ In particular, 



71^ = (e^ 



D^y 



is generated by e ^ . 



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". In 

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

 is the second power of the transformation e^^ of V", and is generated by 

 the infinitesimal transformation e ^^ 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. 



