Solving the generalized Sylvester matrix equation AV+BW=EVF via a Kronecker map

This note considers the solution to the generalized Sylvester matrix equation AV + BW = V F with F being an arbitrary matrix, where V and W are the matrices to be determined. With the help of the Kronecker map, an explicit parametric solution to this matrix equation is established. The proposed solution possesses a very simple and neat form, and allows the matrix F to be undetermined.