## On positive definite solutions of the family of matrix equations $$X+A^*X^{-n}A=Q$$.(English)Zbl 1096.15003

The author studies the matrix equation $$X+A^*X^{-n}A=Q$$ (properties of its maximal and minimal positive definite solutions) and the corresponding matrix function $G(X)=\root n\of{A(Q-X)^{-1}A^*}.$ He gives sufficient conditions for the existence of minimal and special positive definite solutions. The special positive definite solution $$X$$ satisfies the condition $$\| X^{-1}\| \leq \frac{n+1}{n}\| Q^{-1}\|$$. Iterative procedures for computing these solutions are discussed as well as conditions for convergence of the procedures.

### MSC:

 15A24 Matrix equations and identities 15A45 Miscellaneous inequalities involving matrices 65F30 Other matrix algorithms (MSC2010)
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### References:

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