TABER. — LINEAR TRANSFORMATIONS. 377 



But (j> is commutative with 12, and consequently with w. Therefore, 



« Q 12 



£~a ^ = ,75 



to (U 



and by the last theorem we may put 



♦-e-*r©+*.)G-*rs+*) 



= co(12 — a)Y' aW )- I (fi + a.Y' a (o) (12 — uY'fiOiy^n + wY'fiw)^- 1 



= W (0-Y tt )- 1 ((2+Y a )(0-Yp)- 1 (0+Yp)a.- 1 , 



in which Y' a , Y'p, and therefore Y a = uY'.w, Yp = wY'^w, are sym- 

 metric. Whence it follows that 



«£ = a)" 1 4> w = (12 - Y,,)- 1 (12 + Y a ) (12 - Yp)-» (12 + Yp) 



is the product of two of Cayley's expressions.* 

 10. If </> is neither symmetric nor orthogonal, let 



© = 12 c£ ; 



whence it follows that = — ©, and 12 ® = 12. 



Let 6' 12 be a polynomial in © satisfying the equation = e ' a ; and 

 let 



* It is not proved that the two factors of <f> are orthogonal. But it can be 

 readily proved that <j> is equal to the product of three matrices given by Cay- 

 ley's expression, each of which is orthogonal; and therefore the sub-group of 

 orthogonal solutions of the equation <pQ<p = ft can be generated by the Cayleyan 

 solutions of this sub-group; or, what is the same thing, by the orthogonal 

 matrices commutative with ft arid of which +1 is not a latent root. 



For, if <f> is orthogonal, we may put (p = <p <f>, where <p is an orthogonal ma- 

 trix of which —1 is not a latent root and <p is a symmetric orthogonal matrix, 

 both <p and </> being polynomials in <p. (See these Proceedings, Vol. XXVIII. 

 p. 219.) If now <£ft0 = ft, then <f> ft <j> = ft, and </> ft = 12 <p . From the last 



equation follows 



ft ft 



Therefore, by (8) 



*=(sr($-r(3+") 



= «(ft-co 2 )-l (ft + or) (ft -wTco)- 1 (ft + (oTsi)*- 1 . 



Therefore, <p = w~ ] <p w is the product of two of Cayley's expressions. More- 

 over, the first factor, namely, (ft — or) - 1 (ft -f- w-j is orthogonal ; and, since <p is 

 orthogonal, the second factor of <p is also orthogonal. Therefore, <p = <p <t> is 

 the product of three solutions of the equation 0ft0 = ft given by Cayley's ex- 

 pression, each of which is orthogonal. 



