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

The matrix equation X+ i=1 m A i * X -1 A i =I is considered, where A i n×n ,i=1,2,,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 {ϕ j (I)} j=0 are positive definite and the sequence {ϕ j (I) -1 } j=0 is uniformly bounded, where ϕ(X)=I- i=1 m A i * X -1 A i , ϕ 1 =ϕ and ϕ j+1 =ϕ(ϕ j ).

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

Two examples illustrate the behaviour of the proposed algorithms.

MSC:
15A24Matrix equations and identities
15B57Hermitian, skew-Hermitian, and related matrices
65F30Other matrix algorithms
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