## Abstract and Applied Analysis

### Solutions and Improved Perturbation Analysis for the Matrix Equation $X-{A}^{*}{X}^{-p}A=Q \left(p>0\right)$

Jing Li

#### Abstract

The nonlinear matrix equation $X-{A}^{\mathrm{*}}{X}^{-p}A=Q$ with $p>\mathrm{0}$ is investigated. We consider two cases of this equation: the case $p\ge \mathrm{1}$ and the case $\mathrm{0} In the case $p\ge \mathrm{1}$, 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 $\mathrm{0}, 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.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 575964, 12 pages.

Dates
First available in Project Euclid: 27 February 2014

https://projecteuclid.org/euclid.aaa/1393511915

Digital Object Identifier
doi:10.1155/2013/575964

Mathematical Reviews number (MathSciNet)
MR3064406

Zentralblatt MATH identifier
1291.15041

#### Citation

Li, Jing. Solutions and Improved Perturbation Analysis for the Matrix Equation $X-{A}^{*}{X}^{-p}A=Q \left(p&gt;0\right)$. Abstr. Appl. Anal. 2013 (2013), Article ID 575964, 12 pages. doi:10.1155/2013/575964. https://projecteuclid.org/euclid.aaa/1393511915

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