Partition of the Spectrum by Hermite Forms and One-Dimensional Spectral Matrix Portraits
S. K. Godunov
Sobolev Institute of Mathematics
Abstract: There exist classes of nonselfadjoint (in general) matrix operators whose concrete eigenvalues of a spectral cluster are ill-conditioned, whereas the invariant subspace is well-conditioned.
In applications, it is convenient to describe properties of such operators in terms of some criteria for spectral dichotomy. It is convenient to divide the spectrum by a series of plane curves depending on a single parameter. The graphical dependence of the criterion for dichotomy on this parameter is naturally regarded as a spectral portrait. Criteria for dichotomy are connected with Hermite forms. (Recall that Hermite forms appeared in 1856 in solving a similar problem studied by Hermite).