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

Abstract

A complex matrix P ∈ C n × n is said to be a generalized reflection if P = P H = P −1. Let P ∈ C n × n and Q ∈ C n × n be two generalized reflection matrices. A complex matrix A ∈ C n × 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 = 1 l A i XB i + ∑ i = 1 l C i YD i = M , ∑ i = 1 l E i XF i + ∑ i = 1 l 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 ∼ , 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) = AX 2 + BX + C = 0 is highlighted. Finally, two numerical examples are presented to support the theoretical results of this paper.