AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 891840 10.1155/2013/891840 891840 Letter to the Editor Comment on “Perturbation Analysis of the Nonlinear Matrix Equation X-i=1mAi*XpiAi=Q http://orcid.org/0000-0002-0076-0056 Berzig Maher 1 Karapınar Erdal 2 El-Sayed Ahmed 1 Tunis College of Sciences and Techniques Tunis University 5 Avenue Taha Hussein, P.O. Box 56, Bab Manara, Tunis Tunisia utunis.rnu.tn 2 Department of Mathematics Atilim University Incek, 06836 Ankara Turkey atilim.edu.tr 2013 9 10 2013 2013 07 08 2013 04 09 2013 2013 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.

We show that the perturbation estimate for the matrix equation X-i=1mAi*XpiAi=Q 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 𝒞n×n the set of n×n complex matrices, by · the spectral norm, and by λmin(M) the minimal eigenvalues of M.

Consider the matrix equation (1)X-i=1mAi*XpiAi=Q, where Ai𝒞n×n for 1im. The existence and uniqueness of its positive definite solution X is proved in . Next, consider the perturbed equation (2)X~-i=1mA~i*X~piA~i=Q~, where 0<pi<1 and A~i and Q~ are small perturbations of Ai and Q, respectively. We assume that X and X~ are solutions of (1) and (2), respectively. Let (3)ΔX=X~-X,ΔQ=Q~-Q,ΔAi=A~i-Ai.

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 [<xref ref-type="bibr" rid="B1">1</xref>, Theorem 5]).

Let (4)β=λmin(Q)+i=1mλmin(Ai*Ai)λminpi(Q),b=β+ΔQ-i=1mpiβpiAi2,s=i=1mβpiΔAi(2Ai+ΔAi). If (5)0<b<2(β-s)b2-4(β-s)(s+ΔQ)0, then (6)X~-XXρi=1mΔAi+ωΔQ, where (7)ρ=2si=1mΔAi(b+b2-4(β-s)(s+ΔQ)),ω=2b+b2-4(β-s)(s+ΔQ).

2. Counterexample

The following counterexample shows that the perturbation estimates in Theorem 1 are not true in general. Consider (8)q=34,m=1,A=12,  A~=A+110,X=1,X~=X+1100. Now, we compute Q and Q~ by using (9)Q=X-A*XqA,Q~=X~-A~*X~qA~, so we get (10)Q=0.75,Q~0.64730. Finally, using (8)–(10), we obtain that the hypothesis of Theorem 1 is satisfied, that is, (11)0<b0.66815<2(β-s)1.69102,b2-4(β-s)(s+ΔQ)0.435350, whereas (12)X~-XX0.01000ρi=1mΔAi+ωΔQ0.00491.

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.

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