Localization Theorems for Nonlinear Eigenvalue Problems

Let $T : \Omega \rightarrow \mathbb{C}^{n \times n}$ be a matrix-valued function that is analytic on some simply connected domain $\Omega \subset \mathbb{C}$. A point $\lambda \in \Omega$ is an eigenvalue if the matrix $T(\lambda)$ is singular. In this paper, we describe new localization results for...

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Bibliographic Details
Published in:SIAM journal on matrix analysis and applications 2013-01, Vol.34 (4), p.1728-1749
Main Authors: Bindel, David, Hood, Amanda
Format: Article
Language:English
Subjects:
Applied mathematics
Eigenvalues
Localization
Matrix
Theorems
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Summary:Let $T : \Omega \rightarrow \mathbb{C}^{n \times n}$ be a matrix-valued function that is analytic on some simply connected domain $\Omega \subset \mathbb{C}$. A point $\lambda \in \Omega$ is an eigenvalue if the matrix $T(\lambda)$ is singular. In this paper, we describe new localization results for nonlinear eigenvalue problems that generalize Gershgorin's theorem, pseudospectral inclusion theorems, and the Bauer--Fike theorem. We use our results to analyze three nonlinear eigenvalue problems: an example from delay differential equations, a problem due to Hadeler, and a quantum resonance computation. [PUBLICATION ABSTRACT]
ISSN:0895-4798
1095-7162
DOI:10.1137/130913651