Affine subspaces of matrices with rank in a range

TL;DR

This paper investigates the maximal dimension of affine subspaces of matrices with ranks within a specified range, unifying previous results on constant and bounded rank matrices and extending them to more general fields.

Contribution

It provides a comprehensive result on the maximal dimension of affine matrix subspaces with rank constraints, generalizing earlier findings and including new cases like antisymmetric and row echelon matrices.

Findings

Unified bounds for affine subspaces with rank in a range

Extension of constant rank results to more general fields

Results on antisymmetric and row echelon matrices

Abstract

The problem of finding the maximal dimension of linear or affine subspaces of matrices whose rank is constant, or bounded below, or bounded above, has attracted many mathematicians from the sixties to the present day. The problem has caught also the attention of algebraic geometers since vector spaces of matrices of constant rank $r$ give rise to vector bundle maps whose images are vector bundles of rank $r$. Moreover there is a link with the so called ``rank metric codes'', since a constant rank $r$ subspace of $K^{n \times n}$ can be viewed as a constant weight $r$ rank metric code; it can be interesting to study also the maximal dimension of the subspaces of $K^{n \times n}$ whose elements have rank in a range $[s,r]$, since such subspaces obviously give rank metric codes with weights in $[s,r]$. In this paper, with the main purpose to get an organic result including the ones on spaces of matrices with constant rank, the ones on spaces of matrices with rank bounded below and the ones on spaces of matrices with rank bounded above and to generalize a previous result on real matrices with constant rank to matrices on a more general field, we study the maximal dimension of affine subspaces of matrices whose rank is between two numbers under mild assumptions on the field. We get also a result on antisymmetric matrices and on matrices in row echelon form.