Eigenvalue perturbation bounds for Hermitian block tridiagonal matrices

Nakatsukasa, Yuji (2011) Eigenvalue perturbation bounds for Hermitian block tridiagonal matrices. [MIMS Preprint]

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Abstract

We derive new perturbation bounds for eigenvalues of Hermitian matrices with block tridiagonal structure. The main message of this paper is that an eigenvalue is insensitive to blockwise perturbation, if it is well-separated from the spectrum of the diagonal blocks nearby the perturbed blocks. Our bound is particularly effective when the matrix is block-diagonally dominant and graded. Our approach is to obtain eigenvalue bounds via bounding eigenvector components, which is based on the observation that an eigenvalue is insensitive to componentwise perturbation if the corresponding eigenvector components are small. We use the same idea to explain two well-known phenomena, one concerning aggressive early deflation used in the symmetric tridiagonal QR algorithm and the other concerning the extremal eigenvalues of Wilkinson matrices.

Item Type: MIMS Preprint
Uncontrolled Keywords: eigenvalue perturbation, Hermitian matrix, block tridiagonal, Wilkinson's matrix, aggressive early deflation
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory
MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User: Yuji Nakatsukasa
Date Deposited: 19 Dec 2011
Last Modified: 08 Nov 2017 18:18
URI: http://eprints.maths.manchester.ac.uk/id/eprint/1685

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