Stable perturbations of nonsymmetric matrices

James V. Burke, Michael L. Overton

Research output: Contribution to journalArticlepeer-review

Abstract

A complex matrix is said to be stable if all its eigenvalues have negative real part. Let J be a Jordan block with zeros on the diagonal and ones on the superdiagonal, and consider analytic matrix perturbations of the form A(ε) = J + εB + O(ε2), where ε is real and positive. A necessary condition on B for the stability of A(ε) on an interval (0, ε0), and a sufficient condition on B for the existence of such a family A(ε), is (i) Re trB ≤ 0; (ii) the sum of the elements on the first subdiagonal of B has nonpositive real part and zero imaginary part; (iii) the sum of the elements on each of the other subdiagonals of B is zero. This result is extended to matrices with any number of nonderogatory eigenvalues on the imaginary axis, and to a stability definition based on the spectral radius. A generalized necessary condition, though not a sufficient condition, applies to arbitrary Jordan structures. The proof of our results uses two important techniques: the Puiseux-Newton diagram and the Arnold normal form. In the non-deragatory case our main results were obtained by Levantovskii in 1980 using a different proof. Practical implications are discussed.

Original languageEnglish (US)
Pages (from-to)249-273
Number of pages25
JournalLinear Algebra and Its Applications
Volume171
Issue numberC
DOIs
StatePublished - Jul 1 1992

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Numerical Analysis
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics

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