Finite algorithms for solving the coupled Sylvester-conjugate matrix equations over reflexive and Hermitian reflexive matrices

Abstract

It is known that solving coupled matrix equations with complex matrices can be very difficult and it is sufficiently complicated. In this work, we propose two iterative algorithms based on the Conjugate Gradient method (CG) for finding the reflexive and Hermitian reflexive solutions of the coupled Sylvester-conjugate matrix equations (including Sylvester and Lyapunov matrix equations as special cases). The iterative algorithms can automatically judge the solvability of the matrix equations over the reflexive and Hermitian reflexive matrices, respectively. When the matrix equations are consistent over reflexive and Hermitian reflexive matrices, for any initial reflexive and Hermitian reflexive matrices, the iterative algorithms can obtain reflexive and Hermitian reflexive solutions within a finite number of iterations in the absence of roundoff errors, respectively. Finally, two numerical examples are presented to illustrate the proposed algorithms.

Keywords:

• reflexive matrix
• Hermitian reflexive matrix
• iterative algorithm
• coupled Sylvester-conjugate matrix equations

Acknowledgements

The author would like to thank the editor and three anonymous reviewers for their constructive comments and suggestions, which have tremendously improved the original manuscript of this paper.

Additional information

Notes on contributors

Masoud Hajarian

Masoud Hajarian was born in Khorramabad, Iran, on 28 March 1984. He graduated from the Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran in 2006. He received his MSc and PhD in numerical linear algebra, both at Amirkabir University of Technology, Tehran, Iran, in 2008 and 2010, respectively. He is currently working as an Assistant Professor with the Department of Mathematics, Faculty of Mathematical Sciences, Shahid Beheshti University, General Campus, Evin, Tehran 19839, Iran. He is also acting as Associate Editor and reviewer for several journals. His research interests include numerical linear algebra, matrix theory, numerical analysis and operational research.