On spaces of matrices containing a nonzero matrix of bounded rank

Dmitry Falikman, Shmuel Friedland and Raphael Loewy

Vol. 207 (2002), No. 1, 157–176

DOI: 10.2140/pjm.2002.207.157

Abstract

Let M_n(ℝ) and S_n(ℝ) be the spaces of n × n real matrices and real symmetric matrices respectively. We continue to study d(n,n− 2, ℝ): The minimal number ℓ such that every ℓ-dimensional subspace of S_n(ℝ) contains a nonzero matrix of rank n − 2 or less. We show that d(4,2, ℝ) = 5 and obtain some upper bounds and monotonicity properties of d(n,n − 2, ℝ). We give upper bounds for the dimensions of n − 1 subspaces (subspaces where every nonzero matrix has rank n − 1) of M_n(ℝ) and S_n(ℝ), which are sharp in many cases. We study the subspaces of M_n(ℝ) and S_n(ℝ) where each nonzero matrix has rank n or n − 1. For a fixed integer q > 1 we find an infinite sequence of n such that any (q+1 2 ) dimensional subspace of S_n(ℝ) contains a nonzero matrix with an eigenvalue of multiplicity at least q.

Milestones

Received: 5 March 2001

Revised: 6 August 2001

Published: 1 November 2002