Some regularity results for the pseudospectral abscissa and pseudospectral radius of a matrix

Mert G̈urb̈uzbalaban, Michael L. Overton

Research output: Contribution to journalArticle

Abstract

The ε-pseudospectral abscissa αε and radius ρε of an n×n matrix are, respectively, the maximal real part and the maximal modulus of points in its ε-pseudospectrum, defined using the spectral norm. It was proved in [A.S. Lewis and C.H.J. Pang, SIAM J. Optim., 19 (2008), pp. 1048-1072] that for fixed ε > 0, αε and ρε are Lipschitz continuous at a matrix A except when αε and ρε are attained at a critical point of the norm of the resolvent (in the nonsmooth sense), and it was conjectured that the points where αε and ρε are attained are not resolvent-critical. We prove this conjecture, which leads to the new result that αε and ρε are Lipschitz continuous, and also establishes the Aubin property with respect to both ε and A of the ε-pseudospectrum for the points z ε ℂ where αε and ρε are attained. Finally, we give a proof showing that the pseudospectrum can never be "pointed outwards."

Original languageEnglish (US)
Pages (from-to)281-285
Number of pages5
JournalSIAM Journal on Optimization
Volume22
Issue number2
DOIs
StatePublished - 2012

Keywords

  • Aubin property
  • Eigenvalue perturbation
  • Lipschitz multifunction
  • Pseudospectral abscissa
  • Pseudospectral radius
  • Pseudospectrum

ASJC Scopus subject areas

  • Software
  • Theoretical Computer Science

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