216 PROCEEDINGS OP THE AMERICAN ACADEMY 



Since — 1 is not a latent root of \p, 



U I = i. 



0j, ( 0A 2 & 



Or, since e 4 is orthogonal, and \e 4 / = e 2 , therefore, 



•A 



2 



= 1. 



Therefore, every real proper orthogonal matrix has among its 

 square roots one or more real proper orthogonal matrices of which 

 — 1 is not a latent root. 



4. Since 



I <A + 1 | 4= o, 

 we may put 



1 + $> 

 in which case Y will be real, and we shall have 



tr Y - - ^^ - y - y - - Y 



1 — tr. i// l_^-i ,/, _ 1 

 1 + tr. x\i ~ 1 + </,-* ' " ^ + 1 

 and also 



1 — Y 



* = r+Y* 



Therefore, we may put 



* - * = (^D > 



for a proper choice of the skew symmetric matrix Y. 



§ 2. Real Improper Orthogonal Matrices. 



5. If $ is a real improper orthogonal matrix of which unity is a 

 latent root of even multiplicity.* it is the negative of a real proper 

 orthogonal matrix ; therefore hv (4) we may put 



for a proper choice of the real skew symmetric matrix Y. 



* This includes the case in which unity is not a latent root of t I> ; the multi- 

 plicity of unity is then zero. 



