An iterative method for solving the generalized coupled Sylvester matrix equations over generalized bisymmetric matrices

The generalized coupled Sylvester matrix equationsAXB+CYD=M,EXF+GYH=N,(including Sylvester and Lyapunov matrix equations as special cases) have numerous applications in control and system theory. An n×nn×n matrix PP is called a symmetric orthogonal matrix if P=PT=P-1P=PT=P-1. A matrix XX is said to be a generalized bisymmetric with respect to PP, if X=XT=PXPX=XT=PXP. This paper presents an iterative algorithm to solve the generalized coupled Sylvester matrix equations over generalized bisymmetric matrix pair [X,Y][X,Y]. The proposed iterative algorithm, automatically determines the solvability of the generalized coupled Sylvester matrix equations over generalized bisymmetric matrix pair. Due to that II (identity matrix) is a symmetric orthogonal matrix, using the proposed iterative algorithm, we can obtain a symmetric solution pair of the generalized coupled Sylvester matrix equations. When the generalized coupled Sylvester matrix equations are consistent over generalized bisymmetric matrix pair [X,Y][X,Y], for any (spacial) initial generalized bisymmetric matrix pair, by proposed iterative algorithm, a generalized bisymmetric solution pair (the least Frobenius norm generalized bisymmetric solution pair) can be obtained within finite iteration steps in the absence of roundoff errors. Moreover, the optimal approximation generalized bisymmetric solution pair to a given generalized bisymmetric matrix pair can be derived by finding the least Frobenius norm generalized bisymmetric solution pair of new generalized coupled Sylvester matrix equations. Finally, a numerical example is given which demonstrates that the introduced iterative algorithm is quite efficient.