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Abstract and Applied Analysis

Volume 2013 (2013), Article ID 891840, 2 pages

http://dx.doi.org/10.1155/2013/891840

## Comment on “Perturbation Analysis of the Nonlinear Matrix Equation ”

^{1}Tunis College of Sciences and Techniques, Tunis University, 5 Avenue Taha Hussein, P.O. Box 56, Bab Manara, Tunis, Tunisia^{2}Department of Mathematics, Atilim University, Incek, 06836 Ankara, Turkey

Received 7 August 2013; Accepted 4 September 2013

Academic Editor: Ahmed El-Sayed

Copyright © 2013 Maher Berzig and Erdal Karapınar. 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 show that the perturbation estimate for the matrix equation due to J. Li, is wrong. Our discussion is supported by a counterexample.

#### 1. Introduction and Preliminaries

The following definitions and the notations are the same as in [1]. We denote by the set of complex matrices, by the spectral norm, and by the minimal eigenvalues of .

Consider the matrix equation where for . The existence and uniqueness of its positive definite solution is proved in [2]. Next, consider the perturbed equation where and and are small perturbations of and , respectively. We assume that and are solutions of (1) and (2), respectively. Let

In [3, 4], some comments on perturbation estimates for particular cases of (1) and (2) have been furnished. In this note, we focus on the following recent result obtained by J. Li.

Theorem 1 (see [1, Theorem 5]). *Let
**
If
**
then
**
where
*

#### 2. Counterexample

The following counterexample shows that the perturbation estimates in Theorem 1 are not true in general. Consider Now, we compute and by using so we get Finally, using (8)–(10), we obtain that the hypothesis of Theorem 1 is satisfied, that is, whereas

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Authors' Contribution

All the authors contributed equally to this work and significantly in writing this paper. All the authors read and approved the final paper.

#### References

- J. Li, “Perturbation analysis of the nonlinear matrix equation $X-{\sum}_{i=1}^{m}{A}_{i}^{*}{X}^{{p}_{i}}{A}_{i}=Q$,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 979832, 11 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - X. Duan, A. Liao, and B. Tang, “On the nonlinear matrix equation $X-{\sum}_{i=1}^{m}{A}_{i}^{*}{X}^{\delta i}{A}_{i}=Q$,”
*Linear Algebra and its Applications*, vol. 429, no. 1, pp. 110–121, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Berzig, “Comment to: perturbation estimates for the nonlinear matrix equation $X-{A}^{*}{X}^{q}A=Q(0<q<1)$ by G. Jia and D. Gao,”
*Journal of Applied Mathematics and Computing*, vol. 41, no. 1-2, pp. 501–503, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - X. Duan and M. Berzig, “A note concerning Gao’s and Zhang’s perturbation results of the matrix equation $X-{A}^{*}{X}^{q}A=I$,”
*Mathematica Numerica Sinica*, vol. 34, no. 4, pp. 447–447, 2013 (Chinese).