On the matrices of given rank in a large subspace

TL;DR

This paper investigates conditions under which a large subspace of matrices is spanned by matrices of a fixed rank, generalizing known results and providing new sufficient conditions based on codimension and rank.

Contribution

It introduces a sufficient condition on the codimension of a subspace for it to be spanned by matrices of a given rank, extending Gerstenhaber's theorem.

Findings

Provides a generalization of Gerstenhaber's theorem.

Establishes a sufficient codimension condition for spanning by rank r matrices.

Extends previous results on matrices of maximal rank in subspaces.

Abstract

Let V be a linear subspace of M_{n,p}(K) with codimension lesser than n, where K is an arbitrary field and n >=p. In a recent work of the author, it was proven that V is always spanned by its rank p matrices unless n=p=2 and K is isomorphic to F_2. Here, we give a sufficient condition on codim V for V to be spanned by its rank r matrices for a given r between 1 and p-1. This involves a generalization of the Gerstenhaber theorem on linear subspaces of nilpotent matrices.