Solutions to the matrix equation AX 􀀀 EXF D BY

An explicit solution to the generalized Sylvester matrix equation AX 􀀀EXF D BY; with the matrix F being a companion matrix, is given. This solution is represented in terms of the R-controllability matrix of .E; A; B/, generalized symmetric operator and a Hankel matrix. Moreover, several equivalent forms of this solution are presented. The obtained results may provide great convenience for many analysis and design problems. A numerical example is used to illustrate the effectiveness of the proposed approach.