Solution to Generalized Sylvester Matrix Equations

An explicit solution to the generalized Sylvester matrix equation AX+BY= EXF is established. This solution is expressed in terms of the R-controllability matrix of (E, A, B), a generalized symmetric operator matrix and an observability matrix. Moreover, based on this solution, solutions to some other matrix equations are also derived. The results may provide great convenience for the analysis and synthesis problems related to these equations.