Convexity and lipschitz behavior of small pseudospectra

J. V. Burke, A. S. Lewis, M. L. Overton

Research output: Contribution to journalArticlepeer-review


The ε-pseudospectrum of a matrix A is the subset of the complex plane consisting of all eigenvalues of complex matrices within a distance ε of A, measured by the operator 2-norm. Given a nonderogatory matrix A0, for small ε > 0, we show that the ε-pseudospectrum of any matrix A near A0 consists of compact convex neighborhoods of the eigenvalues of A0. Furthermore, the dependence of each of these neighborhoods on A is Lipschitz.

Original languageEnglish (US)
Pages (from-to)586-595
Number of pages10
JournalSIAM Journal on Matrix Analysis and Applications
Issue number2
StatePublished - 2007


  • Eigenvalue optimization
  • Lipschitz multi-function
  • Nonsmooth analysis
  • Pseudospectrum
  • Robust optimization

ASJC Scopus subject areas

  • Analysis


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