TY - JOUR

T1 - On the Lidskii-Vishik-Lyusternik perturbation theory for eigenvalues of matrices with arbitrary Jordan structure

AU - Moro, Julio

AU - Burke, James V.

AU - Overton, Michael L.

N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 1997/10

Y1 - 1997/10

N2 - Let A be a complex matrix with arbitrary Jordan structure and λ an eigenvalue of A whose largest Jordan block has size n. We review previous results due to Lidskii [U.S.S.R. Comput. Math. and Math. Phys., 1 (1965), pp. 73-85], showing that the splitting of λ under a small perturbation of A of order ε is, generically, of order ε1/n. Explicit formulas for the leading coefficients are obtained, involving the perturbation matrix and the eigenvectors of A. We also present an alternative proof of Lidskii's main theorem, based on the use of the Newton diagram. This approach clarifies certain difficulties which arise in the nongeneric case and leads, in some situations, to the extension of Lidskii's results. These results suggest a new notion of Hölder condition number for multiple eigenvalues, depending only on the associated left and right eigenvectors, appropriately normalized, not on the Jordan vectors.

AB - Let A be a complex matrix with arbitrary Jordan structure and λ an eigenvalue of A whose largest Jordan block has size n. We review previous results due to Lidskii [U.S.S.R. Comput. Math. and Math. Phys., 1 (1965), pp. 73-85], showing that the splitting of λ under a small perturbation of A of order ε is, generically, of order ε1/n. Explicit formulas for the leading coefficients are obtained, involving the perturbation matrix and the eigenvectors of A. We also present an alternative proof of Lidskii's main theorem, based on the use of the Newton diagram. This approach clarifies certain difficulties which arise in the nongeneric case and leads, in some situations, to the extension of Lidskii's results. These results suggest a new notion of Hölder condition number for multiple eigenvalues, depending only on the associated left and right eigenvectors, appropriately normalized, not on the Jordan vectors.

KW - Newton diagram

KW - Newton envelope

KW - Perturbation of eigenvalues

KW - Perturbation theory for linear operators

KW - Spectral condition number

KW - Stability theory

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U2 - 10.1137/S0895479895294666

DO - 10.1137/S0895479895294666

M3 - Article

AN - SCOPUS:0031497208

VL - 18

SP - 793

EP - 817

JO - SIAM Journal on Matrix Analysis and Applications

JF - SIAM Journal on Matrix Analysis and Applications

SN - 0895-4798

IS - 4

ER -