## Abstract and Applied Analysis

### Positive Definite Solutions of the Matrix Equation ${X}^{r}-{\sum }_{i=\mathrm{1}}^{m}{A}_{i}^{\mathrm{\ast }}{X}^{-{\delta }_{i}}{A}_{i}=I$

Asmaa M. Al-Dubiban

#### Abstract

We investigate the nonlinear matrix equation ${X}^{r}-{\sum }_{i=\mathrm{1}}^{m}{A}_{i}^{\mathrm{\ast }}{X}^{-{\delta }_{i}}{A}_{i}=I$, where $r$ is a positive integer and ${\delta }_{i}\in (0,1], \text{for}\text{\hspace\{0.17em\}}i=1,2,\dots ,m$. We establish necessary and sufficient conditions for the existence of positive definite solutions of this equation. A sufficient condition for the equation to have a unique positive definite solution is established. An iterative algorithm is provided to compute the positive definite solutions for the equation and error estimate. Finally, some numerical examples are given to show the effectiveness and convergence of this algorithm.

#### Article information

Source
Abstr. Appl. Anal., Volume 2015 (2015), Article ID 473965, 8 pages.

Dates
First available in Project Euclid: 17 August 2015

https://projecteuclid.org/euclid.aaa/1439816232

Digital Object Identifier
doi:10.1155/2015/473965

Mathematical Reviews number (MathSciNet)
MR3372884

Zentralblatt MATH identifier
1314.65062

#### Citation

Al-Dubiban, Asmaa M. Positive Definite Solutions of the Matrix Equation ${X}^{r}-{\sum }_{i=\mathrm{1}}^{m}{A}_{i}^{\mathrm{\ast }}{X}^{-{\delta }_{i}}{A}_{i}=I$. Abstr. Appl. Anal. 2015 (2015), Article ID 473965, 8 pages. doi:10.1155/2015/473965. https://projecteuclid.org/euclid.aaa/1439816232

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