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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 575964, 12 pages
http://dx.doi.org/10.1155/2013/575964
Research Article

Solutions and Improved Perturbation Analysis for the Matrix Equation

School of Mathematics and Statistics, Shandong University, Weihai, Weihai 264209, China

Received 16 November 2012; Revised 2 May 2013; Accepted 7 May 2013

Academic Editor: Carlos Vazquez

Copyright © 2013 Jing Li. 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

The nonlinear matrix equation with is investigated. We consider two cases of this equation: the case and the case In the case , a new sufficient condition for the existence of a unique positive definite solution for the matrix equation is obtained. A perturbation estimate for the positive definite solution is derived. Explicit expressions of the condition number for the positive definite solution are given. In the case , a new sharper perturbation bound for the unique positive definite solution is derived. A new backward error of an approximate solution to the unique positive definite solution is obtained. The theoretical results are illustrated by numerical examples.