218 PEOCEEDINGS OF THE AMERICAN ACADEMY 



Note on Imaginary Orthogonal Matrices. 



Let cf) be an imaginary proper orthogonal matrix whose distinct 

 latent roots are 1, — 1, g, g~ l ; and let the rational integral function 

 of (f> of lowest order that vanishes be 



(<£ - 1)'" (<£ + 1) O - g? (0 - g~'y. 

 Let 



J ^ [(^- 1 ) m -(- 1 - 1 ) m ][(^- 1 )" i -(.y- 1 )"?[(^- 1 )" i -(^~ 1 - 1 )"? 

 [-(-]- 1)-] [- (g- 1)-J» [- (<r 1 - l)™]* 



[(^+l)-(l + l)]" i [(^+l)-(,y+l)?[(^+l)-(r 1 +l)? 

 [-(l + l)] m [-fl'+l)?[-(^- 1 +l)? 



r = [(0-y) p -(i -s^]" 1 [(4>-.y) ?i - (- 1 -<7) p ] [(*-gO' , -(r 1 -g) , y 



[- (1 - <7) p ] m [" (- 1 - #) p ] [- (r 1 ~ 9) P 1 P 



let D be obtained from C by interchanging g and y -1 in the expres- 

 sion for C. Then 



A + B + C+ D = 1; 



.4 2 = A, B' 2 = B, C- = 0, D 1 = B, 



AB=BA = AC=CA = .... = 0, 



(that is, all binary products formed from two different letters vanish) ; 



tr. A = A, tr. B = B, tr. C = D, tr. D = 0. 



Moreover, 



(^-l^-MtO, (<£— l) m ^4 = 0, 



f> + 1) 5 = 0, 



(^-r'^o, y>- 9 yo= o, 



(<£ - (T 1 )"- 1 2? * 0, O- r/" 1 )" 2) = 0. 



I find that the matrix 2? is separable into a sum of two matrices, 

 2?! and 2? 2 , such that 



B? = 2? x , Bi = B 2 , 



tr. 2? 2 = B u 



B X B 2 = B,B, = 0* 



* The products of B x and B 2 by or into either of the letters A, C, or D also 



vanisli. Thus, 



(Bi + B 2 )A = BA = 0; 



.-. B X A = B 1 {B 1 + B 2 )A = 0. 



