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=EVFAV+BW=EVF with FF being an arbitrary matrix, where VV and WW are the matrices to be determined. With the help of the Kronecker map, some properties of the Sylvester sum are first proposed. By applying the Sylvester sum as tools, an explicit parametric solution to this matrix equation is established. The proposed solution is expressed by the Sylvester sum, and allows the matrix FF to be undetermined.