On matrix equations \(X\pm A^*X^{-2}A=I\). (English) Zbl 0979.15007

Solutions of the matrix equations \(X+A^*X^{-2} A=I\) and \(X-A^*X^{-2} A=I\) are studied \((I\) is the \(n\times n\) unit matrix and \(A\) is an \(n\times n\) invertible matrix). Iterative methods are used to find positive definite solutions of these equations. Sufficient conditions are found for the existence of two different solutions of the former equation and sufficient conditions are derived for the existence of positive definite solutions of the latter equations. The equations arise in dynamic programming, stochastic filtering, control theory, and statistics. Numerical examples are discussed.


15A24 Matrix equations and identities
65F10 Iterative numerical methods for linear systems
65F30 Other matrix algorithms (MSC2010)
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