Comment on “Perturbation Analysis of the Nonlinear Matrix Equation 𝑋−∑^𝑚_𝑖=1𝐴^∗_𝑖𝑋^𝑝𝑖𝐴_𝑖=𝑄”

Abstract

We show that the perturbation estimate for the matrix equation 𝑋−∑^𝑚_𝑖=1𝐴^∗_𝑖𝑋^𝑝𝑖𝐴_𝑖=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 [1]. We denote by 𝒞^𝑛×𝑛 the set of 𝑛×𝑛 complex matrices, by ‖⋅‖ the spectral norm, and by 𝜆_min(𝑀) the minimal eigenvalues of 𝑀.

Consider the matrix equation 𝑋−^𝑚_𝑖=1𝐴^∗_𝑖𝑋^𝑝𝑖𝐴_𝑖=𝑄, (1) where 𝐴_𝑖∈𝒞^𝑛×𝑛 for 1≤𝑖≤𝑚. The existence and uniqueness of its positive definite solution 𝑋 is proved in [2]. Next, consider the perturbed equation 𝑋−^𝑚_𝑖=1𝐴^∗_𝑖𝑋^𝑝𝑖𝐴_𝑖=𝑄, (2) where 0<𝑝_𝑖<1 and 𝐴_𝑖 and 𝑄 are small perturbations of 𝐴_𝑖 and 𝑄, respectively. We assume that 𝑋 and 𝑋 are solutions of (1) and (2), respectively. Let Δ𝑋=𝑋−𝑋,Δ𝑄=𝑄−𝑄,Δ𝐴_𝑖=𝐴_𝑖−𝐴_𝑖. (3)

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 𝛽=𝜆_min(𝑄)+^𝑚_𝑖=1𝜆_min𝐴^∗_𝑖𝐴_𝑖𝜆^𝑝𝑖_min(𝑄),𝑏=𝛽+‖Δ𝑄‖−^𝑚_𝑖=1𝑝_𝑖𝛽^𝑝𝑖𝐴_𝑖²,𝑠=^𝑚_𝑖=1𝛽^𝑝𝑖Δ𝐴_𝑖2𝐴_𝑖+Δ𝐴_𝑖. (4) If 𝑏0<𝑏<2𝛽−𝑠₂−4𝛽−𝑠(𝑠+‖Δ𝑄‖)≥0, (5) then 𝑋−𝑋‖𝑋‖≤𝜌^𝑚_𝑖=1Δ𝐴_𝑖+𝜔‖Δ𝑄‖, (6) where 𝜌=2𝑠∑^𝑚_𝑖=1Δ𝐴_𝑖𝑏+𝑏²(‖,2−4𝛽−𝑠𝑠+Δ𝑄‖)𝜔=𝑏+𝑏²(‖.−4𝛽−𝑠𝑠+Δ𝑄‖) (7)

2. Counterexample

The following counterexample shows that the perturbation estimates in Theorem 1 are not true in general. Consider 3𝑞=41,𝑚=1,𝐴=2,1𝐴=𝐴+110,𝑋=1,𝑋=𝑋+.100 (8) Now, we compute 𝑄 and 𝑄 by using 𝑄=𝑋−𝐴^∗𝑋^𝑞𝐴,𝑄=𝑋−𝐴^∗𝑋^𝑞𝐴, (9) so we get 𝑄=0.75,𝑄≈0.64730. (10) Finally, using (8)–(10), we obtain that the hypothesis of Theorem 1 is satisfied, that is, 𝑏0<𝑏≈0.66815<2𝛽−𝑠≈1.69102,₂−4𝛽−𝑠(𝑠+‖Δ𝑄‖)≈0.43535≥0, (11) whereas 𝑋−𝑋‖𝑋‖≈0.01000≰𝜌^𝑚_𝑖=1Δ𝐴_𝑖+𝜔‖Δ𝑄‖≈0.00491. (12)

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.