TABER. — LINEAR TRANSFORMATIONS. 373 



in which 12' = — 17', and 12' 2 = — 1. If ij/ is given by the product of 

 four such expressions as 



in which Y' is symmetric ; then since this expression is equal to 



== co (17 — co Y'tu)- 1 CO . CO" 1 (12 + co Y'co) co" 1 

 = co-(0- Y)- 1 ^^ Y).co-\ 



in which Y = coY' co is symmetric; therefore </> = co - 1 \p co is the pro- 

 duct of four of Cayley's expressions. 



Further, if ^ is a solution of the equation <££&<£ = Q, given by 

 Cayley's expression, then i/^ = co c/> co - ! is a solution of the equation 

 ip 12' {{/ = f2' given by Cayley's expression. Therefore, if if/ is equal to 

 the product into itself an infinite number of times of a solution of the 

 equation ^12'^ = 17', differing iufinitesimally from the matrix unity, 

 cf> is equal to the product into itself an infinite number of times of a 

 solution of the equation ^fi^fi, differing infinitesimally from the 

 matrix unity.* 



3. Let now c/> be any solution of the equation <f> 12 4> = Q, in which 

 il = —17, and 17 2 = —1 (and therefore 12 O = 1). The complete de- 

 termination of <f> may be obtained by the consideration of the identity 



e -ae ne 9(i _ q^ 



oo (0 nv 



in which e en denotes the exponential series 2 r - — p- , convergent for 



any matrix. Thus, let — 17 ^ be a polynomial in c/>, satisfying the 

 equation </> = e -ne . "We then have c/> = e en ; but the identity gives 

 <f> = e en . Since — 170 is a polynomial in c/>, #£2 is a polynomial in c/>; 

 and since c/> -1 = Or 1 <f>Q, $0, = Q- '(17 0)12 is also a polynomial in <£. 

 Consequently, 017 and 0f2 are commutative. Therefore, 



e (B-9)a = e en e -en _ e 0fi.( e en)-i _ ^-i = i. 



* Moreover, if (p is a solution of the equation <p Cl<p = XI, given by Cayley's 

 expression, it may be given the exponential representation e en in which is 

 symmetric. For any symmetric matrix 0, e 6il is a solution of the equation 



0fl<£ = n. Therefore, for any integer n, e" n is a solution. But (e" e )" = e — <& 

 Whence the theorem. See note to (•">). 



