Spaces of matrices with few eigenvalues (II)

TL;DR

This paper extends the classification of large subspaces of matrices with limited eigenvalues to fields of characteristic 2, completing previous work for other characteristics.

Contribution

It determines the maximum dimension of such matrix subspaces over fields with characteristic 2, filling a gap in the existing theory.

Findings

Classified spaces of matrices with at most two eigenvalues over characteristic 2 fields.

Extended previous results to include characteristic 2 fields.

Provided a complete characterization of maximal subspaces in this setting.

Abstract

Let $F$ be a field, and $\mathcal{M}$ be a linear subspace of $n$-by-$n$ matrices with entries in $F$ that have at most two eigenvalues in $F$ (respectively, at most one non-zero eigenvalue in $F$). In a previous article, we have determined the greatest possible dimension for $\mathcal{M}$ when the characteristic of $F$ is not $2$. In this article and its sequel, we solve this problem for all fields with characteristic $2$.