Mathematical Problems in Engineering
Volume 2009 (2009), Article ID 672695, 13 pages
doi:10.1155/2009/672695
Research Article

Necessary and Sufficient Conditions for the Existence of a Hermitian Positive Definite Solution of a Type of Nonlinear Matrix Equations

Wenling Zhao,1 Hongkui Li,1 Xueting Liu,2 and Fuyi Xu1

1College of Science, Shandong University of Technology, Zibo 255049, Shandong, China
2School of Electrical and Electronic Engineering, Shandong University of Technology, Zibo 255049, Shandong, China

Received 13 February 2009; Revised 5 June 2009; Accepted 31 July 2009

Academic Editor: Victoria Vampa

Copyright © 2009 Wenling Zhao 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

We study the Hermitian positive definite solutions of the nonlinear matrix equation X+A∗X−2A=I, where A is an n×n nonsingular matrix. Some necessary and sufficient conditions for the existence of a Hermitian positive definite solution of this equation are given. However, based on the necessary and sufficient conditions, some properties and the equivalent equations of X+A∗X−2A=I are presented while the matrix equation has a Hermitian positive definite solution.