Two iterative algorithms for solving coupled matrix equations over reflexive and anti-reflexive matrices

An n x n real matrix P is said to be a generalized reflection matrix if P-T = P and P-2 = I (where P-T is the transpose of P). A matrix A is an element of R-nxn is said to be a reflexive (anti-reflexive) matrix with respect to the generalized reflection matrix P if A = PAP (A = -PAP). The reflexive and anti-reflexive matrices have wide applications in many fields. In this article, two iterative algorithms are proposed to solve the coupled matrix equations {A(1)XB(1) + (C1XD1)-D-T = M-1, A(2)XB(2) + (C2XD2)-D-T = M-2, over reflexive and anti-reflexive matrices, respectively. We prove that the first (second) algorithm converges to the reflexive (anti-reflexive) solution of the coupled matrix equations for any initial reflexive (anti-reflexive) matrix. Finally two numerical examples are used to illustrate the efficiency of the proposed algorithms.