Affine subspaces of antisymmetric matrices with constant rank

Abstract

For every $n\in \mathbb{N}$ and every field $K$, let $A(n,K)$ be the vector space of the antisymmetric $(n\times n)$-matrices over $K$. We say that an affine subspace $S$ of $A(n,K)$ has constant rank $r$ if every matrix of $S$ has rank $r$. Define $${\cal A}_{antisym}^K(n;r)= \{ S \;| \; S \; \mbox{\rm affine subspace of $A(n,K)$ of constant rank } r\}$$ $$a_{antisym}^K(n;r)=\max \{\dim S\mid S\in {\mathcal{A}}_{antisym}^K(n;r)\}.$$ In this paper we prove the following formulas: for $n\ge 2r+2$ $$a_{antisym}^{\mathbb{R}}(n;2r)=(n-r-1)r;$$ for $n=2r$ $$a_{antisym}^{\mathbb{R}}(n;2r)=r(r-1);$$ for $n=2r+1$ $$a_{antisym}^{\mathbb{R}}(n;2r)=r(r+1).$$

References in corpus (1)

• Affine subspace of matrices with constant rank