TABER. — SCALAR FUNCTIONS OF IIYPEK COMPLEX NUMBERS, GGl 



and, therefore, if we put 



(84) vuv''^ = ^,u<') (i = 1,2,... m; u, v =1,2,... 7i), 



we then have 



(So) /::',■= fr /•:.-22 (,•= 1,2, ...m): 



whence it follows that E'l, £'■_>, • ■ £'m i^i't? linearly independent, and 



(86) E'i E'j = tr. Ei ■ tr. Ej = tr. {Ej Ei) 



mm m 



= tr.iZ yiikEk)= Z yjiktr.Ek= Z VikE'k 



k=l k=l A=l 



(i>j = 1,2, ... m); 



that is to say, the numbers c'l, e'2, ■ ■ ■ e'^ of the quadrate are then 

 linearly independent, and 



m 



(87) e'le'j = Y. yjikc'k {i, j = 1,2, . . . m). 



^-=l 



We may take n = m, and, at the same time, put 



(88) ^„/^') = 7u«, r7.,(') = 7.U. (?, u,v= 1,2, ... m), 

 unless, for Oi, oo, . . . «« not all zero, we ha\'e, simultaneously, 



m m 



(89) 2: tti'Yi.u = Z «i^««^''^ = 0. 



{u,v = 1,2, ... m), 



in which case, neither the m matrices Ei, E2, ■ ■ ■ Ej^ of order m repre- 

 senting, respectively, ei, co, ... e^ nor the m matrices E'], E'2, . . . E'^ 

 representing, respectively, c'l, e'2, . . . e'^, are linearly independent. ^^ 



22 I here follow Cayley in denoting by tr. M the transverse (or conjugate) of 

 any given matrix M. Loo. cit., p. 31. 



23 If n = m and duv^^^ = yivu for i, u, v = \,2, . . . m, the constituent of 

 EiEj in the Mth row and i^th column is 



n n m m 



2 duw'^'Hm^J^ = T. yiwujjvw = ::: tovTutu = 2 inwdJ'^'^ 



»=! w=l w=l w=\ 



by (54); and, therefore, 



m 



EiEj = X JijwE^. 



