On the dimension of some real, bounded rank, matrix spaces

TL;DR

This paper investigates the maximal dimension of certain matrix spaces with high rank, using K-theory and bundle maps to derive sharp estimates for real vector subspaces of hermitian or real matrices.

Contribution

It provides the first sharp bounds on the dimension of real subspaces within high-rank matrix sets, employing K-theory and bundle map techniques.

Findings

Sharp estimate for maximal dimension when n is even

Use of K-theory to analyze matrix rank spaces

Results apply to hermitian and real matrices of rank at least n-1

Abstract

Given $n$ integer, let $X$ be either the set of hermitian or real $n\times n$ matrices of rank at least $n-1$. If $n$ is even, we give a sharp estimate on the maximal dimension of a real vector subspace of $X\cup\{0\}$. The rusults are obtained, via K-theory, by studying a bundle map induced by the adjugation of matrices