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Perturbation theory of selfadjoint matrices and sign characteristics under generic structured rank one perturbations

Please always quote using this URN:urn:nbn:de:0296-matheon-6710
  • For selfadjoint matrices in an indefinite inner product, possible canonical forms are identified that arise when the matrix is subjected to a selfadjoint generic rank one perturbation. Genericity is understood in the sense of algebraic geometry. Special attention is paid to the perturbation behavior of the sign characteristic. Typically, under such a perturbation, for every given eigenvalue, the largest Jordan block of the eigenvalue is destroyed and (in case the eigenvalue is real) all other Jordan blocks keep their sign characteristic. The new eigenvalues, i.e., those eigenvalues of the perturbed matrix that are not eigenvalues of the original matrix, are typically simple, and in some cases information is provided about their sign characteristic (if the new eigenvalue is real). The main results are proved by using the well known canonical forms of selfadjoint matrices in an indefinite inner product, a version of the Brunovsky canonical form and on general results concerning rank one perturbations.

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Metadaten
Author:Christian Mehl, Volker Mehrmann, Andre Ran, Leiba Rodman
URN:urn:nbn:de:0296-matheon-6710
Referee:Günter M. Ziegler
Document Type:Preprint, Research Center Matheon
Language:English
Date of first Publication:2009/10/12
Release Date:2009/09/12
Tag:
Institute:Technische Universität Berlin
MSC-Classification:15-XX LINEAR AND MULTILINEAR ALGEBRA; MATRIX THEORY / 15Axx Basic linear algebra / 15A63 Quadratic and bilinear forms, inner products [See mainly 11Exx]
15-XX LINEAR AND MULTILINEAR ALGEBRA; MATRIX THEORY / 15Bxx Special matrices / 15B57 Hermitian, skew-Hermitian, and related matrices
47-XX OPERATOR THEORY / 47Axx General theory of linear operators / 47A55 Perturbation theory [See also 47H14, 58J37, 70H09, 81Q15]
47-XX OPERATOR THEORY / 47Bxx Special classes of linear operators / 47B50 Operators on spaces with an indefinite metric [See also 46C50]
Preprint Number:675
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