An explicit solution to the matrix equation AX − XF = BY

A complete, general and explicit solution to the generalized Sylvester matrix equation AX − XF = BY, with the matrix F in a companion form, is proposed. The solution is in an extremely neat form represented by a symmetric operator matrix, a Hankel matrix and the controllability matrix of the matrix pair (A, B). Furthermore, several equivalent forms of this solution are also presented. Based on these presented results, explicit solutions to the normal Sylvester equation and the well-known Lyapunov matrix equation are also established. The results provide great convenience to the analysis of the solution to the equation, and can perform important functions in many analysis and design problems in control systems theory. As a demonstration, a simple and effective approach for parametric pole assignment is proposed.