On Extremal Ranks and Least Squares Solutions Subject to a Rank Restriction

Abstract

We discuss the feasible interval of the parameter $k$ and a general expression of matrix $X$ which satisfies the rank equation $r(A-BXC)=k$. With these results, we study two problems under the rank constraint $r(A-BXC)=k$. The first one is to determine the maximal and minimal ranks under the rank constraint $r(A-BXC)=k$. The second one is to derive the least squares solutions of ${\parallel BXC-A\parallel }_F=\min$ under the rank constraint $r(A-BXC)=k$.