Journal of Applied Mathematics
Volume 2012 (2012), Article ID 492951, 20 pages
http://dx.doi.org/10.1155/2012/492951
Research Article

An Iterative Algorithm for the Generalized Reflexive Solution of the Matrix Equations 𝐴 𝑋 𝐵 = 𝐸 , 𝐶 𝑋 𝐷 = 𝐹

Deqin Chen,1 Feng Yin,1 and Guang-Xin Huang2

1School of Science, Sichuan University of Science and Engineering, Zigong 643000, China
2College of Management Science, Key Laboratory of Geomathematics in Sichuan, Chengdu University of Technology, Chengdu 610059, China

Received 21 December 2011; Revised 30 April 2012; Accepted 16 May 2012

Academic Editor: Jinyun Yuan

Copyright © 2012 Deqin Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

An iterative algorithm is constructed to solve the linear matrix equation pair 𝐴 𝑋 𝐵 = 𝐸 , 𝐶 𝑋 𝐷 = 𝐹 over generalized reflexive matrix 𝑋 . When the matrix equation pair 𝐴 𝑋 𝐵 = 𝐸 , 𝐶 𝑋 𝐷 = 𝐹 is consistent over generalized reflexive matrix 𝑋 , for any generalized reflexive initial iterative matrix 𝑋 1 , the generalized reflexive solution can be obtained by the iterative algorithm within finite iterative steps in the absence of round-off errors. The unique least-norm generalized reflexive iterative solution of the matrix equation pair can be derived when an appropriate initial iterative matrix is chosen. Furthermore, the optimal approximate solution of 𝐴 𝑋 𝐵 = 𝐸 , 𝐶 𝑋 𝐷 = 𝐹 for a given generalized reflexive matrix 𝑋 0 can be derived by finding the least-norm generalized reflexive solution of a new corresponding matrix equation pair 𝐴 𝐹 𝑋 𝐵 = 𝐸 , 𝐶 𝑋 𝐷 = with 𝐸 = 𝐸 − 𝐴 𝑋 0 𝐵 , 𝐹 = 𝐹 − 𝐶 𝑋 0 𝐷 . Finally, several numerical examples are given to illustrate that our iterative algorithm is effective.