Applications of linear transformations to matrix equations (Q1373320)

Consider the linear matrix equation \(T(X)=R\), \(T(X): =AX-CXB\), where \(A,C\in \mathbb{C}^{n \times n}\), \(B\in\mathbb{C}^{s\times s}\), \(R\in \mathbb{C}^{n \times s}\) are given matrices such that \(AC=CA\) and \(X\in\mathbb{C}^{n \times s}\) is the unknown matrix. The following is done in this paper: (i) The properties of the linear transformation \(T\) are studied; (ii) An explicit expression for \(X\) is given; (iii) Conditions are derived for the existence of a unique full rank solution \(X\); (iv) An answer is given to the problem: Given the matrices \(A,C,R\) find a matrix \(B\) with a prescribed spectrum such that the solution \(X\) is a full rank matrix.