On affine spaces of rectangular matrices with constant rank

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.