Variational principles for indefinite eigenvalue problems
type: Item
instance of: scholarly article
title: Variational principles for indefinite eigenvalue problems
zbMATH Open document ID: 0821.15005
DOI: 10.1016/0024-3795(93)00187-5
author: Qiang Ye
author: P. A. Binding
published in: Linear Algebra and its Applications
publication date: 1995-09-28T00:00:00Z
review text: The history of the development of variational principles for the selfadjoint generalized eigenvalue problem \(Ax = \lambda Bx\) \((x \neq 0)\) is retrospected in some detail firstly. Restricting \(A\) and \(B\) to be \(n \times n\) complex Hermitian matrices with \(B\) nonsingular but neither \(A\) nor \(B\) definite, the authors generalize all the cited results for the matrix case with semisimple real eigenvalues, allowing nonreal or non- semisimple ones. Let \(S_ k\) denote the linear subspaces of \(\mathbb{C}^ N\) of codimension \(k\) and \(\sigma^ \pm_ k = \text{sup} \{\inf \{x^* Ax : x^* Bx = \pm 1,\;x \in S\} : S \in S_{k - 1}\}\). They show how to calculate the value of each \(\sigma^ 2_ k\) and the corresponding inf sups and in consequence, indicate precisely which eigenvalues of \(Ax = \lambda Bx\) can be obtained by variational formulae of the above types. A simple algorithm is also given for testing exactly which eigenvalues (including those of the mixed type) of this equation can be characterized by \(\sigma^ \pm_ k\) or \(\delta^ \pm_ k\), some of these characterizations being new. This algorithm is illustrated on an example, which shows some of the differences between the cited results.
reviewed by: Yang Yingchen
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 65F15
zbMATH DE Number: 753873
zbMATH Keywords: indefinite eigenvalue problems
zbMATH Keywords: variational principles
zbMATH Keywords: selfadjoint generalized eigenvalue problem
zbMATH Keywords: complex Hermitian matrices
zbMATH Keywords: algorithm
MaRDI profile type: Publication
full work available at URL: 0024-3795(93)00187-5
OpenAlex ID: W2094148697
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description: scientific article; zbMATH DE number 753873