On the matrix equation \(AX=B\) with applications to the generators of a controllability matrix
The following results are obtained concerning the rank of the solutions of the matrix equation \(AX=B\) where A is \(m\times n\) and B is \(m\times p\). 1. There exist solutions of all ranks q with rank \(B\leq q\leq \min (n+rank B-rank A,p)\). 2. If there exists a solution \(X^*\) with rank \(X^*=\ell\) then there exist solutions of all ranks q with \(\ell -r\leq q\leq \min (p,\ell +k-r)\) where \(k=\dim \ker A\) and \(r=\dim (\ker A\cap range X)\). These results are applied to determine when a given \(n\times nm\) matrix \({\mathcal A}\) occurs as a controllability matrix of a pair (A,B).
scientific article; zbMATH DE number 3989561