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{dimS|S∈AK(m×n;r)}.asymK(n;r)=max{dimS|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.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.