OF ARTS AND SCIENCES. 221 



Note on Symmetric Orthogonal Matrices. 



Every symmetric orthogonal matrix is a symmetric square root of 

 unity, and therefore, if 4> is a symmetric orthogonal matrix, an or- 

 thogonal matrix sr can always be found to satisfy the equation 



in which w is a matrix whose constituents are all zero except those in 

 the principal diagonal which are severally equal to ±1. 



If <I? is real, cr may be taken real, and hence it follows from the 

 theorems of § 1 and § 2 that, for a proper choice of the real skew sym- 

 metric matrix Y, and of a>' a matrix similar to w, we may put 



- = (r+^) w 



Therefore, 



