On positive definite solutions of the family of matrix equations \(X+A^*X^{-n}A=Q\). (Q2493935)
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| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On positive definite solutions of the family of matrix equations \(X+A^*X^{-n}A=Q\). |
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On positive definite solutions of the family of matrix equations \(X+A^*X^{-n}A=Q\). (English)
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16 June 2006
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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.
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nonlinear matrix equation
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positive definite solution
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iterative method
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inequalities involving matrices
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convergence
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