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 | References | Similar Articles | Additional Information

Abstract: When properly ordered, the respective eigenvalues of an Hermitian matrix and of a nearby non-Hermitian matrix cannot differ by more than ; moreover, for all , examples and 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 . 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 were normal.

**[A]**Alston S. Householder,*The theory of matrices in numerical analysis*, Blaisdell Publishing Co. Ginn and Co. New York-Toronto-London, 1964. MR**0175290****[A]**A. J. Hoffman and H. W. Wielandt,*The variation of the spectrum of a normal matrix*, Duke Math. J.**20**(1953), 37–39. MR**0052379****[W]**Kahan (1967),*Inclusion theorems for clusters of eigenvalues of Hermitian matrices*, Computer Science Department, University of Toronto, Toronto, Ontario.**[W]**W. Kahan,*Every 𝑛×𝑛 matrix 𝑍 with real spectrum satisfies \Vert𝑍-𝑍*|≤|𝑍+𝑍*\Vert(log₂𝑛+0.038)*, Proc. Amer. Math. Soc.**39**(1973), 235–241. MR**0313278**, https://doi.org/10.1090/S0002-9939-1973-0313278-3**[H]**Weyl (1911),*Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichunger*,.., Math. Ann.**71**, 441-479.**[J]**J. H. Wilkinson,*The algebraic eigenvalue problem*, Clarendon Press, Oxford, 1965. MR**0184422**

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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