Solutions to generalized Sylvester matrix equation by Schur decomposition

This note deals with the problem of solving the generalized Sylvester matrix equation AV-EVF=BW, with F being an arbitrary matrix, and provides complete general parametric expressions for the matrices V and W satisfying this equation. The primary feature of this approach is that the matrix F is firstly transformed into triangular form by Schur decomposition and then unimodular transformation or singular value decomposition are employed. The results can be easily extended to second order case and high order case and can provide great convenience to the computation and analysis of the solutions to this class of equations, and can perform important functions in many analysis and design problems in control systems theory.