Efficient iterative method for solving the second-order Sylvester matrix equation EVF2-AVF-CV=BW

The second-order Sylvester matrix equation EVF2 - AVF - CV = BW (including the generalised Sylvester matrix equation, normal Sylvester matrix equation and Lyapunov matrix equation as special cases) over unknown matrix pair [V, W], has wide applications in many fields. In the present study, the authors propose an iterative method to solve the second-order Sylvester matrix equation. The proposed iterative method does not depend on the Jordan form of the matrix F. By this iterative method, the solvability of the matrix equation can be determined automatically over unknown matrix pair [V, W] not equal 0. When the matrix equation is solvable, its solution pair can be obtained within finite iterative steps, and its least Frobenius norm solution pair can be obtained by choosing suitable initial matrix pair. Furthermore, its optimal approximation solution pair to a given matrix pair can be derived by finding the least norm solution pair of a new matrix equation. A numerical example is given to show the efficiency of the proposed method.