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