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.

Authors