## Abstract and Applied Analysis

### Existence and Uniqueness of the Positive Definite Solution for the Matrix Equation $X=Q+{A}^{\ast }{(\stackrel{^}{X}-C)}^{-1}A$

Dongjie Gao

#### Abstract

We consider the nonlinear matrix equation $X=Q+{A}^{\ast }{(\stackrel{^}{X}-C)}^{-1}A$, where $Q$ is positive definite, $C$ is positive semidefinite, and $\stackrel{^}{X}$ is the block diagonal matrix defined by $\stackrel{^}{X}=\text{d}\text{i}\text{a}\text{g}(X,X,\dots ,X)$. We prove that the equation has a unique positive definite solution via variable replacement and fixed point theorem. The basic fixed point iteration for the equation is given.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 216035, 4 pages.

Dates
First available in Project Euclid: 26 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393447687

Digital Object Identifier
doi:10.1155/2013/216035

Mathematical Reviews number (MathSciNet)
MR3095349

Zentralblatt MATH identifier
1291.15038

#### Citation

Gao, Dongjie. Existence and Uniqueness of the Positive Definite Solution for the Matrix Equation $X=Q+{A}^{\ast }{(\stackrel{^}{X}-C)}^{-1}A$. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 216035, 4 pages. doi:10.1155/2013/216035. https://projecteuclid.org/euclid.aaa/1393447687

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