Spectra of nearly Hermitian matrices

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.

Locations

• Proceedings of the American Mathematical Society - View - PDF