Partition of the spectrum by Hermite forms and one-dimensional spectral matrix portraits

There exist classes of (in general) nonselfadjoint matrix operators whose eigenvalues of a spectral cluster are ill-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 a criterion for dichotomy on this parameter is naturally regarded as 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).