ON DOUBLE POLYADICS — THE LINEAR MATRIX EQUATION. 359 



We have now to develop this determinant by the elements of the 

 first column, yielding the identity 



C M + CM : Ai + C 2 M : A 2 -\ hCjtt : A n = (15) 



where the scalar Co and the polyadics Ci • • • C„ are the cofactors of the 

 elements into which they are multiplied. Evidently Co is the determi- 

 nant of the n 2 elements 0;jt. To exhibit the character of Ci, Co, etc., 

 let Cik denote the cofactor of au- from the n-rowed determinant Co. 

 On developing (14) it appears that 



Ci = - (caEx + c i2 E 2 -\ f- c in E n ) (16) 



whence the C t - are polyadics whose elements are the negatives of the 

 n — 1 rowed cofactors from Co. 



So far the n + 1 polyadics M, Ai, A 2 , • • •, A,, have been quite arbi- 

 trary. If, however, the A t - are linearly independent, their determinant 

 Co is not zero and we may introduce the important new set of polyadics 

 A'i, A' 2 • • • , A'„ defined by 



or by (16) 



C A' t =-Ci (17) 



A\- = i (c a Ei + c, 2 E 2 + • • • + CinE.). (18) 



Co 



The set A'i is said to be reciprocal to the set A;. 



By multiplying the expansions (11) and (18), remembering that 

 Cik is the cofactor of o t ; : from the determinant Co, we have the relation 

 A'i'.Ak = 1 when subscripts are equal, otherwise zero, which we may 

 also express by saying that a set of n polyadics and its reciprocal set 

 with respect to multiple dot product are bi-orthogonal. 



The d may now be eliminated by the aid of (17); and the funda- 

 mental identity (15), (on the hypothesis that Co does not vanish), 

 rearranged in the form 



M = (A'iA, + A' 2 A 2 H 1- A' n A n ) : M. (19) 



The expression in parentheses will be denoted by /; it is an idem factor 

 in the sense that its A'-tuple dot product into an arbitrary Jv-adic 

 leaves that /v-adic unaltered; and it is the most important special 

 case of a double polyadic, — a dyadic whose antecedents and consequents 

 are polyadics. 



Let the expression in parentheses be now transformed by putting 

 in the values of all the polyadics A and A' from the expansions (11) 



