ON DOUBLE POLYADICS — THE LINEAR MATRIX EQUATION. 383 



(32) would still hold if single dot be written for colon. As before re- 

 marked, the star product is essentially double. 



For further comparison, suppose N = 2, with the notation of Art. 10. 



(AM, BN)*s = (on?»22— «i2W2i — a 2 i?»i2+a22Wn) (& u » 22 — b v vri2\~ &21W12 



+ 622" n) 



— (OllW22 — «12»21 — «21»12 +«22^1l) (&11?»22 — &12W21 



— &21W12 + 622WI11) (92) 



which, like (64), reduces to 24 scalar terms by expanding. 



The method of cubic determinants might be applied to star scalars, 

 but with less ease, because a system of dyadics orthogonal under star 

 product is in general imaginary. 



Identities like (44), (42), (45), (47), (48), and (54) to (61) inclusive 

 may be proved in quite parallel steps, with appropriate change of 

 notation. With exception of (54), (56), and (57), these identities do 

 not imply the presence of V — 1 even when we change to star product. 

 While the theory of invariants lies outside the scope of this paper, it 

 may be noted that star scalars, like those of (36), are by nature of their 

 definition, invariant under transformation of the fundamental units, 

 and some of them are invariant in a much wider sense. 



14. Identities with the Star Idemfactor. 



If we have a system of linearly independent dyadics Ai, A 2 , • • -A,, 

 their reciprocals with respect to star product may be called A"i, 

 A" 2 , • • • , A",. By steps as in Art. (3) we may show that 2AA" is 

 identically the idemfactor 2FF. It is evident by Art. 12 that the star 

 idemfactor is real. Taking N — 3 with unit vectors i, j, k, it is easy 

 to verify that the star idemfactor 



- ii(jj + kk - ii) - (ijji + jiij) 



7"= 2 



(93) 



because /"* ii = ii, I"* ij = ij, etc. In fact since, as operators, 



7"* = 7: (94) 



both being identical operations, we must have 



AV= A' r : (95) 



where A" r and A' r are respectively the star and dot reciprocals of A r 

 in the set of linearly independent dyadics Ai, A 2 ,- • •, A„. Operating 

 with both sides of (95) on the dot idemfactor gives 



