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

Abstract

An n × n real matrix P is said to be a generalized reflection matrix if PT = P and P2 = I (where PT is the transpose of P). A matrix A ∈ Rn×n is said to be a reflexive (antireflexive)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 matrixequations over reflexive and anti-reflexive matrices, respectively. We prove that the first (second) algorithm converges to the reflexive (anti-reflexive) solution of the coupled matrixequations for any initial reflexive (anti-reflexive) matrix. Finally two numerical examples are used to illustrate the efficiency of the proposed algorithms. © 2012 SBMAC.