Affine subspace of matrices with constant rank

TL;DR

This paper characterizes the maximum dimensions of affine subspaces of matrices with constant rank, providing explicit formulas for symmetric and general matrices over the real numbers.

Contribution

It derives exact formulas for the maximum dimension of affine subspaces of matrices with constant rank, including symmetric cases, over the real numbers.

Findings

Maximum dimension of affine subspaces of matrices with constant rank over real numbers.

Explicit formula for the maximum dimension in the general case: r(n−r)+r(r−1)/2.

Upper bound for symmetric matrices: floor(r/2)*(n−floor(r/2)).

Abstract

For every $m,n \in \mathbb{N}$ and every field $K$, let $M(m \times n, K)$ be the vector space of the $(m \times n)$-matrices over $K$ and let $S(n,K)$ be the vector space of the symmetric $(n \times n)$-matrices over $K$. We say that an affine subspace $S$ of $M(m \times n, K)$ or of $S(n,K)$ has constant rank $r$ if every matrix of $S$ has rank $r$. Define $${\cal A}^K(m \times n; r)= \{ S \;| \; S \; \mbox{\rm affine subsapce of $M(m \times n, K)$ of constant rank } r\}$$ $${\cal A}_{sym}^K(n;r)= \{ S \;| \; S \; \mbox{\rm affine subsapce of $S(n,K)$ of constant rank } r\}$$ $$a^K(m \times n;r) = \max \{\dim S \mid S \in {\cal A}^K(m \times n; r ) \}.$$ $$a_{sym}^K(n;r) = \max \{\dim S \mid S \in {\cal A}_{sym}^K(n,r) \}.$$ In this paper we prove the following two formulas for $r \leq m \leq n$: $$a_{sym}^{\mathbb{R}}(n;r) \leq \left\lfloor \frac{r}{2} \right\rfloor \left(n- \left\lfloor \frac{r}{2} \right\rfloor\right)$$ $$a^{\mathbb{R}}(m \times n;r) = r(n-r)+ \frac{r(r-1)}{2} .$$