Letter to the Editor | Open Access
Comment on “Perturbation Analysis of the Nonlinear Matrix Equation ”
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
Theorem 1 (see [1, Theorem 5]). Let If then where
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.
All the authors contributed equally to this work and significantly in writing this paper. All the authors read and approved the final paper.
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- X. Duan, A. Liao, and B. Tang, “On the nonlinear matrix equation ,” Linear Algebra and its Applications, vol. 429, no. 1, pp. 110–121, 2008.
- M. Berzig, “Comment to: perturbation estimates for the nonlinear matrix equation by G. Jia and D. Gao,” Journal of Applied Mathematics and Computing, vol. 41, no. 1-2, pp. 501–503, 2013.
- X. Duan and M. Berzig, “A note concerning Gao’s and Zhang’s perturbation results of the matrix equation ,” Mathematica Numerica Sinica, vol. 34, no. 4, pp. 447–447, 2013 (Chinese).
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.