Solution to the generalised sylvester matrix equation AV + BW = AVF

An explicit solution to the generalised Sylvester matrix equation AV+BW=EVF, with the matrix F being in an arbitrary form, is proposed. The solution is in a neat form represented by a generalised symmetric operator matrix, the index-t R-controllability matrix associated with the matrix triple (E, A, B) and an index-t observability matrix. The presented approach allows the matrix F to be set undetermined. This solution can offer all the degrees of freedom. The result may give convenience and advantages to the computation and analysis of the solution to this type of matrix equations. A numerical example is given to illustrate the effect of the proposed approach.