Applications of linear transformations to matrix equations

We consider the linear transformation T(X) = AX − CXB where A, C Mn, B Ms. We show a new approach to obtaining conditions for the existence and uniqueness of the solution X of the matrix equation T(X) = R. As a consequence of our approach we present a simple characterization of a full-rank solution to the matrix equation. We apply the existence theorem to a general form of the observer matrix equation and characterize the existence of a full-rank solution.