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On Hermitian positive definite solutions of matrix equation $X+{A}^{*}{X}^{-2}A=I$. (English) Zbl 1035.15017
The author considers the matrix equation $X+{A}^{*}{X}^{-2}A=I\phantom{\rule{3.33333pt}{0ex}}\left(1\right)$ and its Hermitian positive definite solutions; here $A$ is an $n×n$ complex matrix and $I$ is the identity matrix of order $n$. He shows that if $A$ is normal (i.e. $A{A}^{*}={A}^{*}A$), then such a solution exists if and only if $\rho \left(A\right)\le 2/3\sqrt{3}$ where $\rho \left(A\right)$ is the spectral radius of $A$. The author discusses in detail the basic fixed point iterations for the equation in the case when $A$ is nonnormal and $\parallel A\parallel \le 2/3\sqrt{3}$ where $\parallel ·\parallel$ stands for the spectral norm for square matrices (i.e. one has $0\le {A}^{*}A\le \left(4/27\right)I$). Some of the results of I.G. Ivanov, V.I. Hasanov and B.V. Minchev [ibid. 326, 27-44 (2001; Zbl 0979.15007)] are improved.
MSC:
 15A24 Matrix equations and identities 65F10 Iterative methods for linear systems 65F30 Other matrix algorithms