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On the Hermitian positive definite solution of the nonlinear matrix equation. (English) Zbl 1204.15023

The matrix equation $X+{\sum }_{i=1}^{m}{A}_{i}^{*}{X}^{-1}{A}_{i}=I$ is considered, where ${A}_{i}\in {ℂ}^{n×n},i=1,2,\cdots ,m$.

Some necessary and/or sufficient conditions are given for the existence of a Hermitian positive definite solution. For instance, Theorem 2.2 states that such solution exists if and only if all the matrices in the sequence ${\left\{{\phi }^{j}\left(I\right)\right\}}_{j=0}^{\infty }$ are positive definite and the sequence ${\left\{{\phi }^{j}{\left(I\right)}^{-1}\right\}}_{j=0}^{\infty }$ is uniformly bounded, where $\phi \left(X\right)=I-{\sum }_{i=1}^{m}{A}_{i}^{*}{X}^{-1}{A}_{i}$, ${\phi }^{1}=\phi$ and ${\phi }^{j+1}=\phi \left({\phi }^{j}\right)$.

Two iterative algorithms are provided to determine the maximal positive definite solution.

Two examples illustrate the behaviour of the proposed algorithms.

##### MSC:
 15A24 Matrix equations and identities 15B57 Hermitian, skew-Hermitian, and related matrices 65F30 Other matrix algorithms