Iterative (R, S)-conjugate solutions to the generalised coupled Sylvester matrix equations

ABSTRACT

For given symmetric orthogonal matrices R, S, i.e. R^T = R, R² = I, S^T = S, S² = I, a matrix $A\in {\mathbb{C}}^{n\times s}$ is termed (R, S)-conjugate matrix if $RAS=\bar{A}$. In this paper, an iterative method is constructed to find the (R, S)-conjugate solutions of the generalised coupled Sylvester matrix equations. The consistency of the considered matrix equations over (R, S)-conjugate matrices is discussed. When the matrix equations have a unique (R, S)-conjugate solution pair, the proposed method is convergent for any initial (R, S)-conjugate matrix pair under a loose restriction on the convergent factor. Moreover, the optimal convergent factor of the presented method is derived. Finally, some numerical examples are given to illustrate the results and effectiveness.

KEYWORDS:

• Iterative method
• (R, S)-conjugate matrix
• coupled Sylvester matrix equations

Acknowledgments

The author would like to thank the referees and editor for their constructive comments and helpful suggestions which would greatly improve this paper.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

This work was supported by Scientific Research Fund of SiChuan Provincial Education Department [grant number 16ZA0220].

Notes on contributors

Sheng-Kun Li

Sheng-Kun Li was born in Sichuan, China in 1977. He received his BS and MS degrees in mathematics from Sichuan University, China, in 1999 and 2003, respectively. He received his PhD degree in mathematics from University of Electronic Science and Technology of China, China, in 2011. He is currently an associate professor in College of Applied Mathematics, Chengdu University of Information Technology, China. His research interests include numerical linear algebra and linear matrix equations.