#### 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.