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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Spectra of nearly Hermitian matrices


Author: W. Kahan
Journal: Proc. Amer. Math. Soc. 48 (1975), 11-17
MSC: Primary 15A42
DOI: https://doi.org/10.1090/S0002-9939-1975-0369394-5
MathSciNet review: 0369394
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Abstract: When properly ordered, the respective eigenvalues of an $ n \times n$ Hermitian matrix $ A$ and of a nearby non-Hermitian matrix $ A + B$ cannot differ by more than $ ({\log _2}n + 2.038)\vert\vert B\vert\vert$; moreover, for all $ n \geq 4$, examples $ A$ and $ B$ exist for which this bound is in excess by at most about a factor 3. This bound is contrasted with other previously published over-estimates that appear to be independent of $ n$. Further, a bound is found, for the sum of the squares of respective differences between the eigenvalues, that resembles the Hoffman-Wielandt bound which would be valid if $ A + B$ were normal.


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DOI: https://doi.org/10.1090/S0002-9939-1975-0369394-5
Keywords: Non-Hermitian perturbation, eigenvalue error bounds, generalized Hoffman-Wielandt theorem, generalized Weyl inequality
Article copyright: © Copyright 1975 American Mathematical Society

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