Sweeping words and the length of a generic vector subspace of M-n (F)

摘要

The main result of this short note is a generic version of Paz' conjecture on the length of generating sets in matrix algebras. Consider a generic g-tuple (A) under bar = (A(1),..., A(g)) of n x n matrices over an infinite field. We show that whenever g(2d) >= n(2), the set of all words of degree 2d in (A) under bar spans the full n x n matrix algebra. Our proofs use generic matrices, are combinatorial and depend on the construction of special kinds of directed multigraphs with few edge-disjoint walks. (C) 2016 Elsevier Inc. All rights reserved.