Closed-form solutions to the matrix equation AX − EXF = BY with F in companion form

Abstract

A closed-form solution to the linear matrix equation AX − EXF = BY with X and Y unknown and matrix F being in a companion form is proposed, and two equivalent forms of this solution are also presented. The results provide great convenience to the computation and analysis of the solutions to this class of equations, and can perform important functions in many analysis and design problems in descriptor system theory. The results proposed here are parallel to and more general than our early work about the linear matrix equation AX − XF = BY .

A combination of Sylvester block sum and block matrix Kronecker map for explicit solutions of Sylvester system of matrix equations

“…Recently, closed form solutions of various forms of matrix equations are developed. Some explicit closed‐form solutions of homogeneous and nonhomogeneous Sylvester conjugate matrix equations are provided, where the solutions are expressed in terms of controllability matrices and observability matrices, see the work by Wu et al A closed‐form solution to the linear matrix equation AX − EXF = BY with X and Y unknown and the matrix F being in a companion form is proposed by Zhou and Duan …”