TABER. — LINEAR TRANSFORMATIONS. 379 



11. If <^> is real, 6 may be so chosen that </> and <f> are real ; and 

 then, if 12 is real, © = 12~~ 1 <£ is real. But then, since is real 

 and skew symmetric, — 1 is not a latent root of ©, which is therefore 

 given by Cayley's expression. 



Whence it follows that any real matrix </> satisfying the equation 

 <£ 12<£ = 12, in which 12 is real, is the product of 12" * into two of 

 Cayley's expressions. But 12 -1 cf> is then also a real solution of this 

 equation, and can therefore be thus expressed. Consequently, is 

 equal to the product of two of Cayley's expressions.* 



In each of the factors of given by Cayley's expression, the sym- 

 metric matrix may be taken real. 



12. These theorems may be extended to the real solutions of the 

 equation 12 = 12, in which 12 is any real skew symmetric matrix 

 whose determinant does not vanish. For if 12 is real, the latent roots 

 of — 12 2 are all positive. Therefore, if w denotes a fourth root of — 12 2 

 expressible in powers of 12 2 , w may be taken real. If now is a real 

 solution of the above equation, we have 



12 , 12 



w ( t >(a — i o}(f>ur 1 = 



2 T' 2 



to to 



in which w w x and -^ are real. Therefore, by the preceding 

 theorem, 



= u> (12 - Y,)" 1 (12 + Y ft ) (12 - Y^)- 1 (12 + Y^) to~\ 



in which Y' a , Y'p, and therefore Y a = to Y' a to, Y^ = to Y'p w, are real 

 symmetric matrices. 



Thus it appears that the group of real solutions of the equation 

 012 = 12 can be generated by the Cayleyan solutions of this group, 

 that is, by the totality of expressions of the form (12 — Y) -1 (12 + Y), 

 in which Y is a real symmetric matrix such that | 12 — Y | £ 0. 



§ 2. Bilinear Form, neither Symmetric nor Alternate. 



1. The problem to determine the most general solution of the equa- 

 tion 12 = 12, in which 12 is neither symmetric nor skew symmetric, 

 may, provided neither ± 1 is a latent root of 1212 -1 , be reduced to the 

 solution of two algebraic equations of the nth degree. 



* See ante, p. 178. 



