Affine subspace of matrices with constant rank

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 $${\cal A}^K(m \times n; r)= \{ S \;| \; S \; \mbox{\rm affine subsapce of $M(m \times n, K)$ of constant rank } r\}$$ $${\cal A}_{sym}^K(n;r)= \{ S \;| \; S \; \mbox{\rm affine subsapce of $S(n,K)$ of constant rank } r\}$$ $$a^K(m\times n;r)=\max \{\dim S\mid S\in {\mathcal{A}}^K(m\times n;r)\}.$$ $$a_{sym}^K(n;r)=\max \{\dim S\mid S\in {\mathcal{A}}_{sym}^K(n,r)\}.$$ In this paper we prove the following two formulas for $r\le m\le n$: $$a_{sym}^{\mathbb{R}}(n;r)\le \lfloor \frac{r}{2}\rfloor \left(n-\lfloor \frac{r}{2}\rfloor \right)$$ $$a^{\mathbb{R}}(m\times n;r)=r(n-r)+\frac{r(r-1)}{2}.$$

Cited by in corpus (1)

• Affine subspaces of antisymmetric matrices with constant rank