ON DOUBLE POLYADICS — THE LINEAR MATRIX EQUATION. 373 



Proof. Let every <p be expanded as in (23). The scalar ((<p\ ■ • , 

 <p n ))s is equal to the polyadic determinant of the M t y as shown in 

 Art. 8. To multiply every tp into a double polyadic d is equivalent to 

 performing the same linear transformation upon every M. But the 

 polyadic determinant is the sum of ordinary determinants whose 

 rows consist of elements of the respective polyadics M. It is well 

 known that when every row of a determinant is affected by the same 

 linear transformation the determinant is merely multiplied by the 

 determinant of the transformation. Thus every square bracket 

 term in the expansion of the polyadic determinant of the M;y is multi- 

 plied by the determinant of d. The theorem is therefore proved. 



As an immediate consequence, any polyadic element (£,:E; on the 

 left of the identity (54) may be replaced by <p,:0:Ey, provided we 

 multiply the right side by the determinant of 9, which may be called 

 m g and is the same as the determinant of the n polyadics 6 : Ei, 6 : E 2 

 ••,0:E n . By a proper choice of the arbitrary double polyadic 6, 

 these n polyadics may be made to be any n polyadics whatever, Ai, • • • , 

 A„. If we agree to write [<p;:Ay] for the polyadic determinant whose 

 elements have the form indicated, and [an] for the determinant (in 

 the ordinary sense), whose elements are the scalar elements of the 

 polyadics Ar -A n resolved according to (11), we shall now have the 

 identity 



((<Pi, <P2, • ■ -,<Pn))s — ~7 — f~ (59) 



[ a ij\ 



assuming the polyadics A, linearly independent. Thus the scalar on 

 the left is invariant of the particular choice of these polyadics. 9 



In a similar manner and with similar notation we mav derive from 

 (56) 



[<Pi ' A ;> ^2 : A y , • • • , <p p : Ay, Ay, Ay, ■ • • , A,-] = [chj] (n—p)! 



((<Pl,<P2,--,<Pp))8 (60) 



and from (57) 



[(#'. A,-)* (Ay)"-*] = [an] p! (n - p)! m p (61) 



In employing this abbreviated notation for polyadic determinants, if 

 there are two subscripts the first always refers to the rows, the second 

 to the columns, as in (59) and (60). Where a series of n polyadics is 

 written in brackets as in (60) and (61) these refer to the rows. We 

 pick out an element of a particular column by giving a particular value 



