Affine subspaces of matrices with constant rank

Abstract

For every m,n∈N and every field K, let M(m×n,K) be the vector space of the (m×n)-matrices over K and let S(n,K) be the vector space of the symmetric (n×n)-matrices over K. We say that an affine subspace S of M(m×n,K) or of S(n,K) has constant rank r if every matrix of S has rank r. DefineAK(m×n;r)={S|Saffine subspace of M(m×n,K) of constant rank r}AsymK(n;r)={S|Saffine subspace of S(n,K)of constant rank r}aK(m×n;r)=max⁡{dim⁡S|S∈AK(m×n;r)}.asymK(n;r)=max⁡{dim⁡S|S∈AsymK(n,r)}.In this paper we prove the following two formulas for r≤m≤n:asymR(n;r)≤⌊r2⌋(n−⌊r2⌋)aR(m×n;r)=r(n−r)+r(r−1)2.