TABER. — LINEAR TRANSFORMATIONS. 371 



XVI. 



ON THE GROUP OF AUTOMORPHIC LINEAR TRANS- 

 FORMATIONS OF A BILINEAR FORM. 



By Henry Taber. 



Presented February 14, 1894. 



§ 1. Alternate Bilinear Form. 



1. In the Philosophical Transactions for 1858, Cayley gave the 

 following identity between two matrices, viz. : — 



(fi - Y) (fi + Y)" 1 n (fi - Y)- 1 (Q + Y) = n. 



From this identity Cayley derives the general solution of the matrical 

 equation </> Q </> = fi, in which f2 is a skew symmetric matrix and </> 

 denotes the transverse of </>. Thus for <£ Cayley gives the expres- 

 sion 



(O-Y)" 1 (12 + Y), 



in which Y is an arbitrary skew symmetric matrix, but such that 



| O - Y 1 + 0. 



From this expression may be derived 



J = (ft-Y) (fi + Y)- 1 ; 



therefore, substituting, the above equation is satisfied identically. 



As shown by Cayley, the solution of this equation is equivalent to 

 the determination of the automorphic linear transformation of the 

 alternate bilinear form 



A A 



(Q) (ar lt ar 2 , . . . ) (tf u y 2 , . . . ), 



the z's and y's being cogrediant. 



Cayley's expression gives every solution with non-vanishing deter- 

 minant of the equation <£ O <£ = fi, except those of which —1 is a 

 latent root (root of the characteristic equation)- If | & | ^ 0' and 



