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

摘要

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 /(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 l~Σ_(i=1)A_iYB_i=C, (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.