Primitive spaces of matrices with upper rank two over the field with two elements

TL;DR

This paper completes the classification of matrix spaces with upper rank two over the field with two elements, addressing cases not covered by previous results for larger fields, and applies these findings to various operator and subspace classifications.

Contribution

It provides the first complete classification of primitive spaces of matrices with upper rank two over , extending known results to this special case and applying them to related algebraic structures.

Findings

Classified 3-dimensional subspaces of M_3() with no non-zero eigenvalues

Classified triples of locally linearly dependent operators over 

Classified 3-dimensional affine spaces within ext{GL}_3()

Abstract

For fields with more than $2$ elements, the classification of the vector spaces of matrices with rank at most $2$ is already known. In this work, we complete that classification for the field $\mathbb{F}_2$. We apply the results to obtain the classification of triples of locally linearly dependent operators over $\mathbb{F}_2$, the classification of the $3$-dimensional subspaces of $\text{M}_3(\mathbb{F}_2)$ in which no matrix has a non-zero eigenvalue, and the classification of the $3$-dimensional affine spaces that are included in the general linear group $\text{GL}_3(\mathbb{F}_2)$.