Explicit solutions to the matrix equation E X F  −  AX  =  C

Abstract

The matrix equation EX F − AX = C is investigated in this study, and three approaches are provided to solve this equation. The first approach is to transform it into a real matrix equation with the help of real representation of complex matrices. In the second approach, the solution is given in terms of characteristic polynomial of a constructed matrix pair. In the third approach, the solution can be neatly expressed in terms of controllability matrices and observability matrices. By specialising the obtained solutions of EX F − AX = C, some new expressions of the generalised Sylvester matrix equations are also provided.

A General Approach to Sylvester-Polynomial-Conjugate Matrix Equations

“…When B = 0 and E = I, this matrix equation becomes the normal Sylvester-conjugate matrix equation; when A = I and B = 0, it becomes the Kalman-Yakubovich-conjugate matrix equation; when A = I and S = 0, it becomes the Yakubovich-conjugate matrix equation. Moreover, when A = I, the matrix Equation ( 1) becomes the nonhomogeneous Yakubovich-conjugate matrix equation X + BY = EXF + S investigated in [11]; when B = 0, the matrix Equation ( 1) becomes the extended Sylvester-conjugate matrix equation AX = EXF + S investigated in [14]; when E = I (and X is interchanged with X), the matrix Equation ( 1) becomes the nonhomogeneous Sylvester-conjugate matrix equation AX + BY = XF + S investigated in [8], Section 3, and furthermore, if S = 0, it becomes the homogeneous Sylvester-conjugate matrix equation AX + BY = XF investigated in [8], Section 2. Hence, Equation (1) unifies many important conjugate versions of the Sylvester matrix equation.…”