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

Julio Moro, James V. Burke, Michael L. Overton

Research output: Contribution to journalArticle

Abstract

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.

Original languageEnglish (US)
Pages (from-to)793-817
Number of pages25
JournalSIAM Journal on Matrix Analysis and Applications
Volume18
Issue number4
DOIs
StatePublished - Oct 1997

Keywords

  • Newton diagram
  • Newton envelope
  • Perturbation of eigenvalues
  • Perturbation theory for linear operators
  • Spectral condition number
  • Stability theory

ASJC Scopus subject areas

  • Analysis

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