# Comment on “Perturbation Analysis of the Nonlinear Matrix EquationX-∑i=1mAi*XpiAi=Q”

Authors: Maher Berzig; Erdal Karapınar
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We show that the perturbation estimate for the matrix equationX-∑i=1mAi*XpiAi=Q due to J. Li, is wrong. Our discussion is supported by a counterexample.


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## Body

## 1. Introduction and Preliminaries

The following definitions and the notations are the same as in [1]. 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 Ai∈𝒞n×n for 1≤i≤m. The existence and uniqueness of its positive definite solution X is proved in [2]. 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 [1, Theorem 5]).
Let(4)β=λmin(Q)+∑i=1m‍λmin(Ai*Ai)λminpi(Q),b=β+∥ΔQ∥-∑i=1m‍piβpi∥Ai∥2,s=∑i=1m‍βpi∥ΔAi∥(2∥Ai∥+∥ΔAi∥).
If
(5)0<b<2(β-s)b2-4(β-s)(s+∥ΔQ∥)≥0,
then
(6)∥X~-X∥∥X∥≤ρ∑i=1m‍∥ΔAi∥+ω∥ΔQ∥,
where
(7)ρ=2s∑i=1m∥ΔAi∥(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 Theorem1 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<b≈0.66815<2(β-s)≈1.69102,b2-4(β-s)(s+∥ΔQ∥)≈0.43535≥0,
whereas
(12)∥X~-X∥∥X∥≈0.01000≰ρ∑i=1m‍∥ΔAi∥+ω∥ΔQ∥≈0.00491.



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*Source: 891840-2013-10-09.xml*