Spectra of nearly Hermitian matrices

Kahan, W.

doi:10.1090/S0002-9939-1975-0369394-5

Spectra of nearly Hermitian matrices
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Proc. Amer. Math. Soc. 48 (1975), 11-17 Request permission

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 $({\log _2}n + 2.038)||B||$; 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

Alston S. Householder, The theory of matrices in numerical analysis, Blaisdell Publishing Co. [Ginn and Co.], New York-Toronto-London, 1964. MR 0175290
A. J. Hoffman and H. W. Wielandt, The variation of the spectrum of a normal matrix, Duke Math. J. 20 (1953), 37–39. MR 52379

Inclusion theorems for clusters of eigenvalues of Hermitian matrices

W. Kahan, Every $n\times n$ matrix $Z$ with real spectrum satisfies $\Vert Z-Z^{\ast }\|\leq \|Z+Z^{\ast } \Vert (\log _{2}n+0.038)$, Proc. Amer. Math. Soc. 39 (1973), 235–241. MR 313278, DOI 10.1090/S0002-9939-1973-0313278-3

Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichunger

71

J. H. Wilkinson, The algebraic eigenvalue problem, Clarendon Press, Oxford, 1965. MR 0184422