Explicit solutions to the matrix equation EXF - AX = C

The matrix equation EX̅F - 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 EX̅F - AX = C, some new expressions of the generalised Sylvester matrix equations are also provided.