TABER. — TRANSFORMATION OP A REAL BILINEAR FORM. 813 



The characteristic equation of a transformation of V of the first kind 

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

 each of the negative roots of this equation are all even ; and therefore, 

 for each negative root of this equation, the elementary divisors (elementar 

 Tlteiler) of the characteristic function corresponding to such root occur 

 in pairs with equal exponent.* 



No transformation of V with negative determinant is of the first kind. 

 Let n = 2, and 



jF = (1, b $ ar lf Xo §y u y 2 ) = (x t y x — x, y 2 ) + b x 2 y x . 

 [0, 1 1 



The form Jp is transformed automorphically if 



(*/, x 2 >) =T(x 1 ,x i ) = (-l, %! x 2 ), 



I 0,-1! 



<*', vfi = r- 1 (yi i a) = (- i) %i , y 2 ). 



i-i,-ii 



We have I T \ = + 1, but T 7 is a transformation of the second kind. 

 Whence it follows, for any value of n, that there are forms jp such that I 7 

 contains transformations of the second kind with positive determinant.f 



By definition no transformation of the second kind is an even power 

 of a transformation of V ; but every transformation ofY'of the second 

 kind is the (2 m + l)' h power, for any odd exponent 2 m + 1, of a trans- 

 formation of T' '. Thus, let T be any transformation of V, and, as before, 

 let Fand Wbe real polynomials in J" satisfying the equation 



7 T = e r+w ^ r ~ 1 . 



As shown above, for any real scalar £, e$ v is a transformation of r' ; and, 

 therefore, so also is e wv ~ x = Te~ r . Consequently 



* For the roots of the characteristic equation of /' are the squares of the roots 

 of the characteristic equation of T; and, if T is real, each imaginary root of the 

 characteristic equation of T is paired with its conjugate imaginary. Compare 

 These Proceedings, 31, 189. 



For the relation between the numbers belonging to the roots of the character- 

 istic equation of a linear transformation T and the exponents of the elementary 

 divisors of the characteristic function of T, see Bull. Am. Math. Soc, 2d series, 

 3, 156. 



t The condition, given on p. 308, sufficient that V shall contain transformations 

 of the second kind, is readily obtained. 



