Affine subspaces of antisymmetric matrices with constant rank

TL;DR

This paper determines the maximum dimension of affine subspaces of antisymmetric matrices over real numbers with a fixed constant rank, providing explicit formulas depending on the matrix size and rank.

Contribution

It derives exact formulas for the maximum dimension of affine subspaces of antisymmetric matrices with constant rank over the real numbers.

Findings

Explicit formulas for maximum dimensions depending on n and r.

Results apply to real antisymmetric matrices with fixed rank.

Provides a complete characterization for the maximum dimension of such subspaces.

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 $${\cal A}_{antisym}^K(n;r)= \{ S \;| \; S \; \mbox{\rm affine subspace of $A(n,K)$ of constant rank } r\}$$ $$a_{antisym}^K(n;r) = \max \{\dim S \mid S \in {\cal A}_{antisym}^K(n;r) \}.$$ In this paper we prove the following formulas: for $n \geq 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 $n=2r+1$ $$a_{antisym}^{\mathbb{R}}( n; 2r) = r(r+1) .$$