On extremal ranks and least squares solutions subject to a rank restriction (Q1724154)

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