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

A complex matrix P is an element of C-nxn is said to be a generalized reflection if P = P-H = P-1. Let P is an element of C-nxn and Q is an element of C-nxn be two generalized reflection matrices. A complex matrix A is an element of C-nxn is called a generalized centro-symmetric with respect to (P; Q), if A = PAQ. It is obvious that any n x n complex matrix is also a generalized centro-symmetric matrix with respect to (1; 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) {Sigma(l)(i=1)A(i)XB(i) + Sigma(i=1CiYDi)-C-l = M, Sigma(i=1EiXFi)-E-l + Sigma(l)(i=1)G(i)YH(i) = 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) over tilde, (Y) over tilde] 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) = AX(2) + BX + C = 0 is highlighted. Finally, two numerical examples are presented to support the theoretical results of this paper. (C) 2011 Elsevier Inc. All rights reserved.