Solutions to the matrix equation AX−EXF=BYAX−EXF=BY

An explicit solution to the generalized Sylvester matrix equation AX−EXF=BYAX−EXF=BY, with the matrix FF being a companion matrix, is given. This solution is represented in terms of the R-controllability matrix of (E,A,B)(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.