Large spaces of symmetric or alternating matrices with bounded rank

TL;DR

This paper classifies large subspaces of symmetric and alternating matrices with bounded rank over arbitrary fields, extending previous results to near-maximal dimensions and removing field size restrictions.

Contribution

It extends the classification of large symmetric matrix spaces with bounded rank to near-maximal dimensions over fields with more than two elements, and similarly for alternating matrices without field restrictions.

Findings

Classified symmetric matrix spaces with near-maximal dimension over large fields.

Established results for alternating matrices with bounded rank over any field.

Generalized Loewy's previous classification results.

Abstract

Let $r$ and $n$ be positive integers such that $r<n$, and $\mathbb{K}$ be an arbitrary field. In a recent work, we have determined the maximal dimension for a linear subspace of $n$ by $n$ symmetric matrices with rank less than or equal to $r$, and we have classified the spaces having that maximal dimension. In this article, provided that $\mathbb{K}$ has more than two elements, we extend this classification to spaces whose dimension is close to the maximal one: this generalizes a result of Loewy. We also prove a similar result on spaces of alternating matrices with bounded rank, with no restriction on the cardinality of the underlying field.