The solution to the matrix equation AV + BW = EVJ + R

Abstract

This note considers the solution to the generalized Sylvester matrix equation AV + BW = EVJ + R, where A, B, E, and R are given matrices of appropriate dimensions, J is an arbitrary given Jordan matrix, while V and W are matrices to be determined. A general parametric solution for this equation is proposed, based on the Smith form reduction of the matrix [ A − sE B]. The solution possesses a very simple and neat form, and does not require the eigenvalues of matrix J to be known. An example is presented to illustrate the proposed solution.