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Preprint 4-2018

First order structure-preserving perturbation theory for eigenvalues of symplectic matrices

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Author(s) : Fredy Sosa, Julio Moro, Christian Mehl

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 4-2018

MSC 2000

15A18 Eigenvalues, singular values, and eigenvectors

Abstract :
A first order perturbation theory for eigenvalues of real or complex J-symplectic matrices under structure- preserving perturbations is developed. As main tools structured canonical forms and Lidskii-like formulas for eigenvalues of multiplicative perturbations are used. Explicit formulas, depending only on appropriately normalized left and right eigenvectors, are obtained for the leading terms of asymptotic expansions describing the perturbed eigenvalues. Special attention is given to eigenvalues on the unit circle, especially to the exceptional eigenvalues 1, whose behavior under structure-preserving perturbations is known to differ significantly from the behavior under general perturbations. Several numerical examples are used to illustrate the asymptotic expansions.

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