On the reflexive and anti-reflexive solutions of the generalised coupled Sylvester matrix equations

Abstract

The generalised coupled Sylvester matrix equations

with unknown matrices X and Y, have important applications in control and system theory. Also it is well known that the reflexive and anti-reflexive matrices have wide applications in many fields. In this article, we consider the generalised coupled Sylvester matrix equations over reflexive and anti-reflexive matrices. First we propose two new matrix equations equivalent to the generalised coupled Sylvester matrix equations over reflexive and anti-reflexive matrices, respectively. Then two new iterative algorithms are proposed for solving these matrix equations. The convergence analysis of the proposed iterative algorithms is derived. Finally, some numerical examples are presented to illustrate the theoretical results of this article.

Keywords:

• generalised coupled Sylvester matrix equations
• reflexive matrix
• anti-reflexive matrix
• iterative algorithm
• Frobenius norm

Acknowledgements

The authors would like to express their heartfelt thanks to the four reviewers and associate editor for their useful comments and constructive suggestions which substantially improved the quality of this article. The authors are also deeply grateful to Professor Peter Fleming (Editor-in-Chief), Professor C.W. Chan (Regional Editor for Asia) for managing the review process.