On affine spaces of rectangular matrices with constant rank

TL;DR

This paper generalizes a result on the maximum dimension of affine subspaces of matrices with constant rank over arbitrary fields, linking it to the classification of quadratic forms.

Contribution

It extends Rubei's real-field result to arbitrary fields with sufficient elements and classifies maximal spaces based on invertible matrix subspaces and quadratic form classifications.

Findings

Generalization of maximal affine subspace dimensions to arbitrary fields.

Classification of spaces reaching maximal dimension based on quadratic forms.

Connection established between matrix space classification and quadratic form theory.

Abstract

Let $\mathbb{F}$ be a field, and $n \geq p \geq r>0$ be integers. In a recent article, Rubei has determined, when $\mathbb{F}$ is the field of real numbers, the greatest possible dimension for an affine subspace of $n$--by--$p$ matrices with entries in $\mathbb{F}$ in which all the elements have rank $r$. In this note, we generalize her result to an arbitrary field with more than $r+1$ elements, and we classify the spaces that reach the maximal dimension as a function of the classification of the affine subspaces of invertible matrices of $\mathrm{M}_s(\mathbb{F})$ with dimension $\dbinom{s}{2}$. The latter is known to be connected to the classification of nonisotropic quadratic forms over $\mathbb{F}$ up to congruence.