The generalised Sylvester matrix equations over the generalised bisymmetric and skew-symmetric matrices

Abstract

A matrix P is called a symmetric orthogonal if P = P ^T  = P ⁻¹. A matrix X is said to be a generalised bisymmetric with respect to P if X = X ^T  = PXP. It is obvious that any symmetric matrix is also a generalised bisymmetric matrix with respect to I (identity matrix). By extending the idea of the Jacobi and the Gauss–Seidel iterations, this article proposes two new iterative methods, respectively, for computing the generalised bisymmetric (containing symmetric solution as a special case) and skew-symmetric solutions of the generalised Sylvester matrix equation

(including Sylvester and Lyapunov matrix equations as special cases) which is encountered in many systems and control applications. When the generalised Sylvester matrix equation has a unique generalised bisymmetric (skew-symmetric) solution, the first (second) iterative method converges to the generalised bisymmetric (skew-symmetric) solution of this matrix equation for any initial generalised bisymmetric (skew-symmetric) matrix. Finally, some numerical results are given to illustrate the effect of the theoretical results.

Keywords:

• generalised Sylvester matrix equation
• iterative method
• convergence
• generalised bisymmetric matrix
• skew-symmetric matrix
• symmetric orthogonal matrix

Acknowledgements

The authors would like to express their heartfelt thanks to two anonymous referees, especially one of them, for very useful comments and constructive suggestions which led to a significant improvement of the quality and presentation of this article. The authors are also very much indebted to the Associate Editor for his valuable comments, generous encouragement and continuous concern during the review process of this article.