Generic eigenstructures of Hermitian pencils

TL;DR

This paper characterizes the generic eigenstructures of complex Hermitian matrix pencils with bounded rank, revealing differences between full-rank and lower-rank cases and highlighting the role of real eigenvalues and sign characteristics.

Contribution

It provides a complete classification of eigenstructures for Hermitian pencils with rank constraints, including explicit counts and codimensions of bundle closures.

Findings

For full-rank Hermitian pencils, eigenstructures can include complex eigenvalues.

For lower-rank pencils, eigenstructures contain only real eigenvalues.

The sign characteristic influences the generic eigenstructure classification.

Abstract

We obtain the generic complete eigenstructures of complex Hermitian $n\times n$ matrix pencils with rank at most $r$ (with $r\leq n$). To do this, we prove that the set of such pencils is the union of a finite number of bundle closures, where each bundle is the set of complex Hermitian $n\times n$ pencils with the same complete eigenstructure (up to the specific values of the finite eigenvalues). We also obtain the explicit number of such bundles and their codimension. The cases $r=n$, corresponding to general Hermitian pencils, and $r<n$ exhibit surprising differences, since for $r<n$ the generic complete eigenstructures can contain only real eigenvalues, while for $r=n$ they can contain real and non-real eigenvalues. Moreover, we will see that the sign characteristic of the real eigenvalues plays a relevant role for determining the generic eigenstructures of Hermitian pencils.