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Notes on two perturbation estimates of the extreme solutions to the equations \(X\pm A^{*}X^{-1}A=Q\). (English) Zbl 1222.15020
Summary: Two perturbation estimates of the maximal positive definite solutions to the matrix equations \(X+A^{*}X^{-1}A=Q\) and \(X-A^{*}X^{-1}A=Q\) are considered. These estimates are like to the estimates discussed by V. I. Hasanov and I. G. Ivanov [Linear Algebra Appl. 413, No. 1, 81–92 (2006; Zbl 1087.15016)]. The conditions \(\| X_L^{-1}A\|_2 < 1\) and \(\| X_+^{-1}A\|_2 < 1\) in [loc. cit.] are not always satisfied. We replace this conditions by \(\| PX_L^{-1}AP^{-1}\|_2 < 1\) and \(\| PX_+^{-1}AP^{-1}\|_2 < 1\) respective, where \(P\) is positive definite matrix. The theoretical results are illustrated by numerical examples.

MSC:
15A24 Matrix equations and identities
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