Matchings and canonical forms for symmetric tensors

Abstract

Let V be a q-dimensional vector space. Fix a set B of q(q - 1) monomials in Sp(V) of the form xI where ik > 0 for all k. The generic element of Sp(V) is conjugate under a suitable linear transformation to an element with support off of B. We prove this by showing the existence of a perfect matching with a unique weight in a certain weighted bipartite graph. Such a perfect matching corresponds to the non-vanishing of an appropriate determinant. © 1996 Academic Press, Inc.