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The Hermitian positive definite solution and its perturbation analysis for the matrix equation $X-A^*X^{-1}A=Q$. (Chinese. English summary) Zbl 1174.65385
Summary: We consider the nonlinear matrix equation $X-A^*X^{-1}A=Q$, where $A$, $Q$ are $n\times n$ complex matrices with $Q$ Hermitian positive definite and $A^*$ denotes the conjugate transpose of the matrix $A$. This paper shows that there exists a unique positive definite solution to the equation. The perturbation bounds for the Hermitian positive definite solution to the matrix equation are derived, explicit expressions of the condition number for the Hermitian positive definite solution are obtained and the backward error of an approximate solution to the Hermitian positive definite solution is evaluated. The results are illustrated by numerical examples.

65F30Other matrix algorithms
15A24Matrix equations and identities