A gradient-based iterative algorithm for generalized coupled Sylvester matrix equations over generalized centro-symmetric matrices

Abstract

An $n \times n$ real matrix $P$ is said to be a symmetric orthogonal matrix if $P = P^{- 1} = P^{T}$ . An $n \times n$ real matrix $X$ is called a generalized centro-symmetric matrix with respect to $P$ , if $X = PXP$ . It is obvious that every $n \times n$ matrix is also a generalized centro-symmetric matrix with respect to $I$ (identity matrix). In the present paper, we propose a gradient-based iterative algorithm to solve the generalized coupled Sylvester matrix equations

${\begin{matrix} A_{1} X_{1} B_{1} + C_{1} X_{2} D_{1} = E_{1}, \\ A_{2} X_{1} B_{2} + C_{2} X_{2} D_{2} = E_{2}, \end{matrix}$

over the generalized centro-symmetric matrix pair $(X_{1}, X_{2})$ . It is proved that the iterative method is always convergent for any initial generalized centro-symmetric matrix pair $(X_{1} (1), X_{2} (1)) .$ Finally, a numerical example is discussed to illustrate the results.

Keywords

Symmetric orthogonal matrix generalized centro-symmetric matrix iterative method the generalized coupled Sylvester matrix equations

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