Analysis of an iterative algorithm to solve the generalized coupled Sylvester matrix equations

A complex matrix P∈Cn×nP∈Cn×n is said to be a generalized reflection if P = PH = P−1. Let P∈Cn×nP∈Cn×n and Q∈Cn×nQ∈Cn×n be two generalized reflection matrices. A complex matrix A∈Cn×nA∈Cn×n is called a generalized centro-symmetric with respect to (P; Q), if A = PAQ. It is obvious that any n × n complex matrix is also a generalized centro-symmetric matrix with respect to (I; I). In this work, we consider the problem of finding a simple way to compute a generalized centro-symmetric solution pair of the generalized coupled Sylvester matrix equations (GCSY)∑i=1lAiXBi+∑i=1lCiYDi=M,∑i=1lEiXFi+∑i=1lGiYHi=N,(including Sylvester and Lyapunov matrix equations as special cases) and to determine solvability of these matrix equations over generalized centro-symmetric matrices. By extending the idea of conjugate gradient (CG) method, we propose an iterative algorithm for solving the generalized coupled Sylvester matrix equations over generalized centro-symmetric matrices. With the iterative algorithm, the solvability of these matrix equations over generalized centro-symmetric matrices can be determined automatically. When the matrix equations are consistent over generalized centro-symmetric matrices, for any (special) initial generalized centro-symmetric matrix pair [X(1), Y  (1)], a generalized centro-symmetric solution pair (the least Frobenius norm generalized centro-symmetric solution pair) can be obtained within finite number of iterations in the absence of roundoff errors. Also, the optimal approximation generalized centro-symmetric solution pair to a given generalized centro-symmetric matrix pair [X∼,Y∼] can be derived by finding the least Frobenius norm generalized centro-symmetric solution pair of new matrix equations. Moreover, the application of the proposed method to find a generalized centro-symmetric solution to the quadratic matrix equation Q(X) = AX2 + BX + C = 0 is highlighted. Finally, two numerical examples are presented to support the theoretical results of this paper.