212 PROCEEDINGS OF THE AMERICAN ACADEMY 



XIII. 



ON REAL ORTHOGONAL SUBSTITUTION. 



By Henry Taber, Clark University. 



Presented April 12, 1893. 



§ 1. Real Proper Orthogonal Matrices. 



L In a paper to appear iu a forthcoming number of the " Quarterly 

 Journal of Mathematics " I have shown that, if cb is any real proper 

 orthogonal matrix, then, for a proper choice of the real skew sym- 

 metric matrix 0, we may put ^ = e^, where e^ denotes the exponential 



0'' . . 

 series ^,.-j , which is convergent for any matrix. This theorem was 



published in these Proceedings, Vol. XXVI. It follows immediately 

 from this theorem that any real orthogonal matrix whatever is given 

 by the expression 



U + Y/ ' 



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

 matrix w whose constituents are all zero except those in the principal 

 diagonal which are severally equal to ± 1. The second factor in the 

 above expression is the square of Cayley's expression for an orthogo- 

 nal matrix. 



If the determinant of the orthogonal matrix is equal to unity, we 

 may put w = 1 ; if the determinant is equal to — 1, and if at the 

 same time unity is a latent root of even multiplicity,* we may put 

 OJ = — 1. 



I shall show in this section that, as a consequence of the exponential 

 representation of real proper orthogonal matrices, any such matrix 

 may be represented by the square of Cayley's expression ; and in § 2 

 it will be shown that the general theorem given above follows as a 

 consequence of the theorem just stated. 



* If unity is not a latent root of the orthogonal matrix, its multiplicity is 

 zero. 



