on double polyadics — the linear matrix equation. 393 



20. The Equation of Extent Two and Order Two. 



The following considerations will suggest how the work of solution 

 may be arranged when the equation is of higher extent. We have 

 seen that any coefficient m p is a sum of groups of terms. Each group 

 is a homogeneous polynomial. Suppose a choice of subscripts in 

 which i occurs a times, j occurs b times, etc. Let the development of 

 terms, under the rule, corresponding to this choice of subscripts, be 

 denoted by G(i a j b - • ■). Each choice of subscripts will occur a number 

 of times equal to the coefficient of the corresponding term in the 

 expansion of 



(i + j+k+---+r + s+ t)*>. (142) 



Thus the entire development may be systematically carried out. 

 For example, take 



AixBi+ A 1 xB j =C (143) 



and form m±. With the above notation we shall have 



m 4 = G^) + iG(Pj) + 6G(W + 46'(y 3 ) + G(j») (144) 



By the preceding example we have G(z 4 ) = A"fsB"h and similarly 

 for G'O' 4 ). By the rule we have 



G(Pj) = A s iS A 3 sB\ s B ]S - SAWAiAda&isiBiBds 



- 3A iS A jS (A%) s B lS B jS (B\) + 3(^)5(^)5(^)5(^)3 

 + QAisiA^A^sBisiB^B^s + 2Aj S (A\) s B jS (B\)s 



- 6 {A*iAi)8(B*&,)a (145) 



Brevity will be gained in notation, while nothing is lost in explicitness, 

 if, in such expressions, we indicate only the matrices A which are 

 pref actors in the original equation, remembering that the order of 

 subscripts among postfactors, when more than two matrices are 

 multiplied, is the reverse of that for prefactors. With this under- 

 standing we may also omit the letter A and the subscript S. Thus 

 (145) may be abbreviated 



G(Pj) = t^-3^(v)-3ii(^)+3(?) (ij)+Wi 2 J)+2J(i 3 )-Gtt s J) (145a) 



As a check, if Aj and B, are replaced by the idemf actor of the second 

 order, G(i 3 j) should reduce to 6AsBsA"sB"s because w 4 becomes the 

 determinant of A( )B + ( ), and the terms of the third degree in A 



