TABER. — TRANSFORMATION OP A REAL BILINEAR FORM. 311 



IV. If Tis real and each negative root of the characteristic equation 

 of 7' is in absolute value less than unity, there is a real polynomial f (2^ 

 satisfying the equation 1 + T = e''^. 



Finally, from the identity, 



e-^^-''Ae'' = Ae-^A-^ • Ae"" = Ae''' e"" = A, 



-ACA~ 



we derive the theorems : 



V. If 6^ satisfies the equation 



(4) U=AUA-'^ 



and 7"= e'^is real, in particular if Cis real, then Tis a transformation 

 of r'.* 



VI. If CT" satisfies the equation 



(5) U=-A UA-^ 



and 7^= e^is real, in particular if U'ls real, then 7^ is a transformation 

 of r". 



VII. If C satisfies the equation 



(6) U=AUA-\ 



and 7^= e^is real, in particular if Cis real, then T'is a transformation 



of r'". 



§2. T-^AT=A. 



If Uh real and satisfies the equation 



(4) U=A UA-\ 



then, by Theorem V, every transformation of the group 6=^, with real 

 parameter ^, is a transformation of T' ; in particular the infinitesimal 

 transformation e*^^ of this group is a transformation of T'. 



If the real infinitesimal transformation e^^^ = 1 + S^U isa. transfor- 

 mation of r', then first U is real ; moreover, 



A + SC(- UA + AU) = (1 - 8^ U) A(l + h^U) = e'^^^'A e^^"" = A , 



and therefore UA ■=■ ATI, that is, U satisfies equation (4j. Therefore, 

 for any real scalar ^, the transformation e^^ generated by e is a trans- 

 formation of r'. Whence it follows that evtry transformation generated 



.. * In this case the application of the above identity is unnecessary; for £/'4,.= 

 A U, and therefore T-^AT=e-^A e^= 4 e'^ e^ = A. 



