OF ARTS AND SCIENCES. 163 



XL 



NOTE ON THE REPRESENTATION OF ORTHOGONAL 



MATRICES. 



By Henry Taber, Clark University. 



Presented May 24, 1892. 



If 4> '^ an orthogonal matrix of which — 1 is not a latent root, 



wt may put 



1 -,p 

 v — - • 



" 1 + 0' 

 whence follows 



tr Y = 1 ~ tr - * = 1 - < ^ 1 = till = _ Y • 

 1 + tr.0 1 + 0- 1 0+1 



and wo have 



1- Y 

 * = f+Y' 



This is Cayley's well known representation of an orthogonal 

 matrix in terms of a skew symmetric matrix. If — 1 is a latent 

 root of <£, hut not 1, — 4> will he an orthogonal matrix of which 

 1 is a latent root, hut not — 1; therefore, — may he represented 

 as above, giving 



Thus the expression 



1- Y 



± r+r 



will, for a proper value of the skew symmetric matrix Y, give all 

 orthogonal matrices except those which have as latent roots both 

 ± 1. Such of these matrices as are real, and of which the multi- 

 jdicity of the latent root — 1 is even, and, more generally, any Teal 

 proper orthogonal matrix, can, I find, be represented as an ex- 

 ponential function of a real skew symmetric matrix. 



