On affine spaces of rectangular matrices with constant rank

2024/05/04 by Clément de Seguins Pazzis, Pazzis, Clément de Seguins · 1 citation
Computer Science · Decision Sciences · Mathematics · #Advanced Topics in Algebra #FOS: Mathematics #Fuzzy and Soft Set Theory #Matrix Theory and Algorithms #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2405.02689

openalex publication_date 2024/05/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \mathbbF be a field, and n ≥ p ≥ r>0 be integers. In a recent article, Rubei has determined, when \mathbbF is the field of real numbers, the greatest possible dimension for an affine subspace of n--by--p matrices with entries in \mathbbF 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 M_s(\mathbbF) with dimension \dbinoms2. The latter is known to be connected to the classification of nonisotropic quadratic forms over \mathbbF up to congruence.

Cited by

• On the dimension of affine subspaces of nilpotent matrices

Related

• On affine spaces of alternating matrices with constant rank
• A dimension bound for constant rank subspaces of matrices over a finite field
• Maximal dimension of affine subspaces of specific matrices
• Affine spaces of symmetric or alternating matrices with bounded rank
• The classification of large spaces of matrices with bounded rank