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

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 Sigma(l)(i=1) A(i)YB(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 (skewsymmetric) 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.