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.


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