On the condition numbers of a multiple eigenvalue of a generalized eigenvalue problem

Nakatsukasa, Yuji (2011) On the condition numbers of a multiple eigenvalue of a generalized eigenvalue problem. [MIMS Preprint]

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Abstract

For standard eigenvalue problems, closed-form expressions for the condition numbers of a multiple eigenvalue are known. In particular, they are uniformly 1 in the Hermitian case and generally take di�erent values in the non-Hermitian case. We consider the generalized eigenvalue problem and identify the condition numbers. Our main result is that a multiple eigenvalue generally has multiple condition numbers, even in the Hermitian de�nite case. The condition numbers are characterized in terms of the singular values of the outer product of the corresponding left and right eigenvectors.

Item Type:	MIMS Preprint
Uncontrolled Keywords:	multiple eigenvalue, generalized eigenvalue problem, condition number
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:	22 Oct 2011
Last Modified:	08 Nov 2017 18:18
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/1689

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