Affine spaces of symmetric or alternating matrices with bounded rank

TL;DR

This paper determines the maximum dimension of affine subspaces of symmetric or alternating matrices with bounded rank over any field, and classifies the subspaces achieving this maximum, extending previous linear case results.

Contribution

It generalizes prior linear subspace results to affine subspaces over arbitrary fields, providing a complete classification of maximal dimension subspaces with bounded rank.

Findings

Maximal dimension of affine subspaces with bounded rank determined

Classification of subspaces up to congruence provided

Results extend previous work to arbitrary fields and affine cases

Abstract

Let $r$ and $n$ be positive integers such that $r<n$, and $\mathbb{K}$ be an arbitrary field. We determine the maximal dimension for an affine subspace of $n$ by $n$ symmetric (or alternating) matrices with entries in $\mathbb{K}$ and with rank less than or equal to $r$. We also classify, up to congruence, the subspaces of maximal dimension among them. This generalizes earlier results of Meshulam, Loewy and Radwan that were previously known only for linear subspaces over fields with large cardinality and characteristic different from $2$.