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## Computational & Applied Mathematics

*On-line version* ISSN 1807-0302

#### Abstract

DEHGHAN, Mehdi and HAJARIAN, Masoud. **Two iterative algorithms for solving coupled matrix equations over reflexive and anti-reflexive matrices**.* Comput. Appl. Math.* [online]. 2012, vol.31, n.2, pp.353-371.
ISSN 1807-0302. http://dx.doi.org/10.1590/S1807-03022012000200008.

An *n × n* real matrix *P* is said to be a generalized reflection matrix if *P ^{T} = P* and

*P*

^{2}=

*I*(where

*P*is the transpose of

^{T}*P*). A matrix A ∈ R

*is said to be a reflexive (anti-reflexive) matrix with respect to the generalized reflection matrix*

^{n×n}*P*if

*A*=

*P A P*(

*A*= -

*P A P*). 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}+

*C*

_{1}

*X*

^{T}D_{1}=

*M*

_{1}.

*A*

_{2}

*XB*2 +

*C*

_{2}

*X*

^{T}D_{2}=

*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.

**Mathematical subject classification:**15A06, 15A24, 65F15, 65F20.

**Keywords
:
**iterative algorithm; matrix equation; reflexive matrix; anti-reflexive matrix.