# Maximal rank in matrix spaces via graph matchings

## TL;DR

This paper investigates the maximum possible rank in affine subspaces of symmetric or alternating matrices by relating it to graph matching numbers, providing new bounds and proofs.

## Contribution

It introduces a novel approach connecting matrix rank problems with graph matchings, offering simplified proofs and bounds for subspace dimensions.

## Key findings

- Established bounds on the dimension of matrix subspaces based on maximal rank.
- Linked matrix rank properties to graph matching numbers.
- Provided simplified proofs for existing bounds.

## Abstract

We study the maximal rank in affine subspaces of symmetric or alternating matrices, in terms of the matching numbers of certain associated graphs. Applications include simple proofs of upper bounds on the dimension of such subspaces in terms of their maximal rank.