## Positive definite solutions of the matrix equations $$X\pm A^{\ast}X^{-q} A=Q$$.(English)Zbl 1078.15012

Nonlinear matrix equations $$X\pm A^*X^{-q} X= Q$$ are studied, where the unknown matrix $$X$$ is assumed to be positive definite. The real parameter $$q$$ satisfies $$0< q\leq 1$$. Criteria for existence of solutions and their upper and lower bounds are established. Sufficient conditions are given for uniqueness of a (positive definite) solution of the equation with the plus sign. For the equation with the minus sign, sufficient conditions are given for existence of two different solutions. Iterative methods are developed.

### MSC:

 15A24 Matrix equations and identities 65F30 Other matrix algorithms (MSC2010)
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### References:

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