Analysis of an iterative algorithm to solve the generalized coupled Sylvester matrix equations (Q638860)

A complex matrix \(P \in \mathbb C^{n \times n}\) is said to be a generalized reflection if \(P=P^H=P^{-1}\). Let \(P \in \mathbb C^{n \times n}\) and \(Q \in \mathbb C^{n \times n}\) be two generalized reflection matrices. A complex matrix \(A \in \mathbb C^{n \times n}\) is called generalized centro-symmetric with respect to \((P;Q)\) if \(A = P A Q\). It is obvious that any \(n \times n\) complex matrix is also a generalized centro-symmetric matrix with respect to \((I;I)\). In this work, the problem is considered of finding a simple way to compute a generalized centro-symmetric solution pair of the generalized coupled Sylvester matrix equations \[ \begin{aligned} &\sum_{i=1}^l A_i X B_i + \sum_{i=1}^l C_i Y D_i = M, \\ &\sum_{i=1}^l E_i X F_i + \sum_{i=1}^l G_i Y H_i = N, \end{aligned} \] (including Sylvester and Lyapunov matrix equations as special cases) and to determine the solvability of these matrix equations over generalized centro-symmetric matrices. By extending the idea of the conjugate gradient method, an iterative algorithm is proposed for solving the generalized coupled Sylvester matrix equations over generalized centro-symmetric matrices. Moreover, the application of the proposed method to find a generalized centro-symmetric solution to the quadratic matrix equation \(Q(X)=A X^2 + B X + C = 0\) is highlighted. Finally, two numerical examples are presented to support the theoretical results.