216 PROCEEDINGS OP THE AMERICAN ACADEMY 



Since — 1 is not a latent root of if/, 



U I = 1. 



Or, since e* is orthogonal, and \e^ J = e^, therefore, 



\ a I I A. 2 



"A 



= 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 I i 0, 

 we may put 



in which case Y will k^e real, and we shall have 



tr Y = ^-^'-^ = l-r^ ^ t^zl = _Y 

 1 + tr.ij/ I + i}/-^ >A + 1 ' 



and also 



1 - Y 



Y 1 + Y 



Therefore, we may put 



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 imity is a 

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

 orthogonal matrix ; therefore by (4) we may put 



* = -(! 



— Y 



+ Y/' 

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



* This includes the case in which unity is not a latent root of "J" ; the multi- 

 plicity of unity is then zero. 



