On the generalized bisymmetric and skew-symmetric solutions of the system of generalized Sylvester matrix equations

Abstract

A matrix P is called a symmetric orthogonal matrix if P = P ^T  = P ⁻¹. A matrix X is said to be a generalized bisymmetric with respect to P, if X = X ^T  = PXP. It is obvious that every symmetric matrix is a generalized bisymmetric matrix with respect to I (identity matrix). In this article, we establish two iterative algorithms for solving the system of generalized Sylvester matrix equations

(including the Sylvester and Lyapunov matrix equations as special cases) over the generalized bisymmetric and skew-symmetric matrices, respectively. When this system is consistent over the generalized bisymmetric (skew-symmetric) matrix Y, firstly it is demonstrated that the first (second) algorithm can obtain a generalized bisymmetric (skew-symmetric) solution for any initial generalized bisymmetric (skew-symmetric) matrix. Secondly, by the first (second) algorithm, we can obtain the least Frobenius norm generalized bisymmetric (skew-symmetric) solution for special initial generalized bisymmetric (skew-symmetric) matrices. Moreover, it is shown that the optimal approximate generalized bisymmetric (skew-symmetric) solution of this system for a given generalized bisymmetric (skew-symmetric) matrix can be derived by finding the least Frobenius norm generalized bisymmetric (skew-symmetric) solution of a new system of generalized Sylvester matrix equations. Finally, the iterative methods are tested with some numerical examples.

Keywords:

• system of generalized Sylvester matrix equations
• generalized bisymmetric matrix
• skew-symmetric matrix
• iterative algorithm

AMS Subject Classifications::

• 15A24
• 65F10
• 65F30

Acknowledgements

The authors are grateful to the anonymous reviewer and Prof. Vadim Olshevsky for their valuable comments and careful reading of the original manuscript of this article. The authors are also very much indebted to Prof. Steve Kirkland (Editor-in-Chief) for his valuable suggestions, generous concern and continuous encouragement during the review process of this article.