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 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]**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 matrix with real spectrum satisfies*, 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