On the generalized bisymmetric and skew-symmetric solutions of the system of generalized Sylvester matrix equations

A matrix P is called a symmetric orthogonal matrix if P = P-T = P-1. A matrix X is said to be a generalized bisymmetric with respect to P, if X = X-T = PXP. It is obvious that every symmetric matrix is a generalized bisymmetric matrix with respect to I (identity matrix). In this article, we establish two iterative algorithms for solving the system of generalized Sylvester matrix equations A(1)YB(1) + A(2)YB(2) + ... + A(l)YB(l) = M, C1YD1 + C2YD2 + ... + ClYDl = N, (including the Sylvester and Lyapunov matrix equations as special cases) over the generalized bisymmetric and skew-symmetric matrices, respectively. When this system is consistent over the generalized bisymmetric (skew-symmetric) matrix Y, firstly it is demonstrated that the first (second) algorithm can obtain a generalized bisymmetric (skew-symmetric) solution for any initial generalized bisymmetric (skew-symmetric) matrix. Secondly, by the first (second) algorithm, we can obtain the least Frobenius norm generalized bisymmetric (skew-symmetric) solution for special initial generalized bisymmetric (skew-symmetric) matrices. Moreover, it is shown that the optimal approximate generalized bisymmetric (skew-symmetric) solution of this system for a given generalized bisymmetric (skew-symmetric) matrix (Y) over cap can be derived by finding the least Frobenius norm generalized bisymmetric (skew-symmetric) solution of a new system of generalized Sylvester matrix equations. Finally, the iterative methods are tested with some numerical examples.