• Ronny Ramlau, Lothar Reichel (Hg.)

ETNA - Electronic Transactions on Numerical Analysis

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Electronic Transactions on Numerical Analysis (ETNA) is an electronic journal for the publication of significant new developments in numerical analysis and scientific computing. Papers of the highest quality that deal with the analysis of algorithms for the solution of continuous models and numerical linear algebra are appropriate for ETNA, as are papers of similar quality that discuss implementation and performance of such algorithms. New algorithms for current or new computer architectures are appropriate provided that they are numerically sound. However, the focus of the publication should be on the algorithm rather than on the architecture. The journal is published by the Kent State University Library in conjunction with the Institute of Computational Mathematics at Kent State University, and in cooperation with the Johann Radon Institute for Computational and Applied Mathematics of the Austrian Academy of Sciences (RICAM). Reviews of all ETNA papers appear in Mathematical Reviews and Zentralblatt für Mathematik. Reference information for ETNA papers also appears in the expanded Science Citation Index. ETNA is registered with the Library of Congress and has ISSN 1068-9613.

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Verlag der Österreichischen Akademie der Wissenschaften
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ETNA - Electronic Transactions on Numerical Analysis



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Verlag der Österreichischen Akademie der Wissenschaften
Austrian Academy of Sciences Press
A-1011 Wien, Dr. Ignaz Seipel-Platz 2,
Tel. +43-1-515 81/DW 3420, Fax +43-1-515 81/DW 3400
https://verlag.oeaw.ac.at, e-mail: bestellung.verlag@oeaw.ac.at
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Relative perturbation tanΘ-theorems for definite matrix pairs

    Suzana Miodragović, Ninoslav Truhar, Ivana Kuzmanović Ivičić

ETNA - Electronic Transactions on Numerical Analysis, pp. 364-380, 2024/07/01

doi: 10.1553/etna_vol60s364

doi: 10.1553/etna_vol60s364


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doi:10.1553/etna_vol60s364



doi:10.1553/etna_vol60s364

Abstract

In this paper, we consider perturbations of a Hermitian matrix pair $(H, M)$, where $H=GJG^*$ is non-singular, $J={\rm diag\,}(\pm 1)$,and $M$ is a positive definite matrix. The corresponding perturbed pair defined as $(\widetilde H, \widetilde M)=(H+\delta H, M+\delta M)$ is such that $\widetilde H=\widetilde G J \widetilde G^*$ is non-singular and $\widetilde M$ is a positive definite matrix. An upper bound for the norm of the tangents of the angles between the eigenspaces of the perturbed and unperturbed pairs is derived. The rotation of the eigenspaces under a perturbation is measured in the scalar product induced by $M$.We show that a relative $\tan\Theta$-bound for the standard eigenvalue problem is a special case of our new bound.

Keywords: perturbation of matrix pairs, rotation of subspaces, tangent theta theorem, eigenvalues, eigenspaces

Published Online: 2024/07/01 09:50:06

Object Identifier: 0xc1aa5572 0x003f332a

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