TABER. — TRANSFORMATION OF A REAL BILINEAR FORM. 311 



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

 of T'\s in absolute value less than unity, there is a real polynomial f (T) 

 satisfying the equation 1 + 7'= e t(T) . 



Finally, from the identity, 



-AUA 



-1 



*— - 1 Ae u =Ae- u A~ l • Ae u = Ae~ u e u = A, 



we derive the theorems : 



V. If U satisfies the equation 



(4) U=AUA~ l 



and T — e^ is real, in particular if U is real, then T is a transformation 



of r'.* 



VI. If U satisfies the equation 



(5) U= - A UA-i 



and T = e n is real, in particular if £7 is real, then T'is a transformation 

 of r". 



VII. If U satisfies the equation 



(6) U=AUA~\ 



and T ■=■ e F is real, in particular if Cis real, then Tis a transformation 



of r'". 



§2. T- X AT=A. 



If £7" is real and satisfies the equation 



(4) U=A UA~\ 



then, by Theorem V, every transformation of the group e^ u , with real 

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

 transformation e ^ u of this group is a transformation of V. 



If the real infinitesimal transformation e 5 ^ = 1 -f- ^C^ is a transfor- 

 mation of T', then first U is real ; moreover, 



A + S£ (- UA + A U) = (1 - 3£ £7) A (1 + 8£tf) = e~^ n A e s < a = A , 



and therefore UA = AU, that is, U satisfies equation (4). Therefore, 

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

 formation of T'. Whence it follows that every transformation generated 



* In this case the application of the above identity is unnecessary ; for U A = 

 A U, and therefore T~ x A T=e~ u A e p =4e _ V = i, 



