A new solution to the generalized Sylvester matrix equation AV-EVF=BWAV-EVF=BW

This note deals with the problem of solving the generalized Sylvester matrix equation AV-EVF=BWAV-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 solution is that the matrix F does not need to be in any canonical form, and may be even unknown a priori. The results 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.