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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 824641, 8 pages
http://dx.doi.org/10.1155/2013/824641
Research Article

A Note on the -Stein Matrix Equation

Center for General Education, National Formosa University, Huwei 632, Taiwan

Received 16 May 2013; Revised 16 July 2013; Accepted 16 July 2013

Academic Editor: Masoud Hajarian

Copyright © 2013 Chun-Yueh Chiang. 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

This note is concerned with the linear matrix equation , where the operator denotes the transpose ( ) of a matrix. The first part of this paper sets forth the necessary and sufficient conditions for the unique solvability of the solution . The second part of this paper aims to provide a comprehensive treatment of the relationship between the theory of the generalized eigenvalue problem and the theory of the linear matrix equation. The final part of this paper starts with a brief review of numerical methods for solving the linear matrix equation. In relation to the computed methods, knowledge of the residual is discussed. An expression related to the backward error of an approximate solution is obtained; it shows that a small backward error implies a small residual. Just like the discussion of linear matrix equations, perturbation bounds for solving the linear matrix equation are also proposed in this work.