218 PROCEEDINGS OP THE AMERICAN ACADEMY 



Note on Imaginary Orthogonal Matrices. 



Let (fi be an imaginary proper orthogonal matrix whose distinct 

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

 of <j) of lowest order that vanishes be 



(^ - 1)- (c^ + 1) (cA - gy (cf. - g-'y. 

 Let 



.•^ [(<^-0"'-(-i-i)"'][(<^-i)'"-(r/-i)"?[(<^-i)"'-(^-^-i)"? 



[-(-]- 1)'«] [- {g- i)"t [- {g-' - 1)"? 



[(c^+l)-(l + l)]'"[(cA+l)-(g^+l)?[(<A+l)-(r^+l)? 



[-(i + i)r[-^+i)?[-(y-^+i)? 

 [- (1 - gyY' [- (- 1 - 9y] [- (^-^ - 9yY 



let Z) be obtained from G by interchanging ^ and g~^ in the expres- 

 sion for C. Then 



^ + 5+ C+ Z) =r 1; 



.42 = J, i?2 ^ B, C- = C, X)2 ^ Z), 

 AB=BA=.AC=GA = ....^0, 



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



tr. A = A, tr.B= B, tr. G = D, tr. D ^ G. 

 Moreover, 



{4> — iy-^Axo, ((^ — i)"'^ = o, 



(.^ + 1) i? = 0, 



(c^-gy-^G^o, (cp-gyG = 0, 



(<^ - 9-'y-' ^ * 0, (<^- ^-1)'' i) - 0. 



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

 Bi and B2, such that 



B,' = B„ B,' = B„ 



tr. B2 = B^, 



BiB, ^ B,B^ = 0* 



* The products of B^ and B2 by or into either of the letters A, C, or D also 



vanish. Thus, 



{B^ + B2)A = BA =0; 



.'. B^A = B^(Bi + B2)A = 0. 



