On varieties of optimal algorithms for the computation of bilinear mappings III. Optimal algorithms for the computation of xy and yx where x, yϵM2(K)

AbstractIn this paper we consider optimal algorithms for the computation of Φ:(x,y)↦ (xy,yx), where x, y are 2×2-matrices over a field K. It is shown that, if the characteristic of K is different from two, then optimal algorithms for the computation of Φ have length nine. Moreover, there are several different equivalence classes of optimal algorithms for Φ. If K= GF(2), then optimal algorithms for the computation of Φ have length ten