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Proceedings of the American Mathematical Society

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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.


References [Enhancements On Off] (What's this?)

  • [A] S. Householder (1964), The theory of matrices in numerical analysis, Blaisdell, New York. MR 30 #5475. MR 0175290 (30:5475)
  • [A] J. Hoffman and H. W. Wielandt (1953), The variation of the spectrum of a normal matrix, Duke Math. J. 20, 37-39. MR 14, 611. MR 0052379 (14:611b)
  • [W] Kahan (1967), Inclusion theorems for clusters of eigenvalues of Hermitian matrices, Computer Science Department, University of Toronto, Toronto, Ontario.
  • [W] Kahan (1973), Every $ n \times n$ matrix $ Z$ with real spectrum satisfies $ \vert\vert Z - {Z^ \ast }\vert\vert \leq \vert\vert Z + {Z^ \ast }\vert\vert({\log _2}n + 0.038)$, Proc. Amer. Math. Soc. 39, 235-241. MR 47 #1833. MR 0313278 (47:1833)
  • [H] Weyl (1911), Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichunger,.., Math. Ann. 71, 441-479.
  • [J] H. Wilkinson (1965), The algebraic eigenvalue problem, Clarendon Press, Oxford. MR 32 #1894. MR 0184422 (32:1894)

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Additional Information

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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