Set-Valued and Variational Analysis

, Volume 22, Issue 1, pp 19-43

First online:

Variational Analysis of the Spectral Abscissa at a Matrix with a Nongeneric Multiple Eigenvalue

  • Sara GrundelAffiliated withComputational Methods in Systems and Control Theory, Max Planck Institute for Dynamics of Complex Technical Systems Email author 
  • , Michael L. OvertonAffiliated withCourant Institute of Mathematical Sciences, New York University

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The spectral abscissa is a fundamental map from the set of complex matrices to the real numbers. Denoted α and defined as the maximum of the real parts of the eigenvalues of a matrix X, it has many applications in stability analysis of dynamical systems. The function α is nonconvex and is non-Lipschitz near matrices with multiple eigenvalues. Variational analysis of this function was presented in Burke and Overton (Math Program 90:317–352, 2001), including a complete characterization of its regular subgradients and necessary conditions which must be satisfied by all its subgradients. A complete characterization of all subgradients of α at a matrix X was also given for the case that all active eigenvalues of X (those whose real part equals α(X)) are nonderogatory (their geometric multiplicity is one) and also for the case that they are all nondefective (their geometric multiplicity equals their algebraic multiplicity). However, necessary and sufficient conditions for all subgradients in all cases remain unknown. In this paper we present necessary and sufficient conditions for the simplest example of a matrix X with a derogatory, defective multiple eigenvalue.


Variational analysis Spectral abscissa Degenerate eigenvalue Derogatory eigenvalue Subgradient

Mathematics Subject Classifications (2010)

49J52 49J53