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Letter to the Editor | Open Access

Volume 2013 |Article ID 891840 | https://doi.org/10.1155/2013/891840

Maher Berzig, Erdal Karapınar, "Comment on “Perturbation Analysis of the Nonlinear Matrix Equation ", Abstract and Applied Analysis, vol. 2013, Article ID 891840, 2 pages, 2013. https://doi.org/10.1155/2013/891840

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

Accepted04 Sep 2013
Published09 Oct 2013

#### 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 . 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 . 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.

1. 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 Site | Google Scholar | MathSciNet
2. 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.
3. M. Berzig, “Comment to: perturbation estimates for the nonlinear matrix equation $X-{A}^{*}{X}^{q}A=Q\left(0 by G. Jia and D. Gao,” Journal of Applied Mathematics and Computing, vol. 41, no. 1-2, pp. 501–503, 2013. View at: Publisher Site | Google Scholar | MathSciNet
4. 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). View at: Google Scholar

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