On the optimal evaluation of a set of bilinear forms

Although general theories are beginning to emerge in the area of automata based complexity theory, there are very few general methods or even general problem formulations in the area of arithmetic complexity. In this paper we propose and defend a general model for studying bilinear multiplication in order to provide a common framework for discussing a wide class of problems. At the heart of a number of problems in minimizing the number of multiplications required to perform a calculation is a problem in matrix algebra relating to the expansion of a given set of matrices as linear combinations of rank one matrices. In this paper we make a systematic attack on this problem and derive some general results which unify and extend numerous known results. Among the new results given here to illustrate the strength of this approach is a new lower bound on the number of multiplications required for n by n matrix multiplication of 3n2-3n+1 which is independent of the subset of the reals with respect to which multiplication is regarded as free. An even sharper bound is obtained if this set is restricted to the integers.