(8) 



TABER. — TRANSFORMATION OF A REAL BILINEAR FORM. 315 



T-^ = A TA-^ ; 



and therefore each root, other than ± 1, of the characteristic equation of 

 7^ is paired with its inverse, so that if ^ =|= =b 1 is a root of this equation, 

 ^., = 4~^ is also a root, and the numbers belonging to ^^ and 4-, (in par- 

 ticular the multiplicity of these roots) are the same. Therefore, by 

 Theorem III, there is a polynomial f{T) satisfying the equation 

 T=e^^^> and such that/(r) —f{T-^) is real. Let now 



2 U, =f(T) +f{T-% 2 IT, =f{T) -/(2-0; 



and let 



To = e\ 71 = e^K 



We have by (8) 



—_ qAT) qAATa'^) 



— g/lD . ^ gAT) J-1 



= TAT- A-^ = A A-'' = 1. 

 Therefore, 



T^=(To T,Y = To"" 7;2 = T,\ 



But, by equation (8), 



= ^(f(AT-^A-^)-f(ATA-^)) 



= iA(f{T-^) -f{T))A-' = -AU,A-'; 



and therefore, since Ui is real, it follows that e^^', for any real scalar ^, 

 is generated by the infinitesimal transformation e^^^i of T". In partic- 

 ular 



T^ = 7\2 = (e^i)2 = g2oi 



.is generated by e^i\ 



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



