Generic symmetric matrix pencils with bounded rank

TL;DR

This paper characterizes the structure and generic eigenstructures of symmetric matrix pencils with bounded rank, revealing their algebraic and geometric properties and identifying their irreducible components.

Contribution

It provides an explicit description of the generic eigenstructures and the algebraic components of symmetric matrix pencils with bounded rank, advancing understanding of their classification.

Findings

The set of symmetric matrix pencils of rank at most r is a union of closures of specific eigenstructure sets.

These closures correspond to the irreducible components of the algebraic set of such pencils.

The identified eigenstructures are the generic complete eigenstructures for these pencils.

Abstract

We show that the set of $n \times n$ complex symmetric matrix pencils of rank at most $r$ is the union of the closures of $\lfloor r/2\rfloor +1$ sets of matrix pencils with some, explicitly described, complete eigenstructures. As a consequence, these are the generic complete eigenstructures of $n \times n$ complex symmetric matrix pencils of rank at most $r$. We also show that these closures correspond to the irreducible components of the set of $n\times n$ symmetric matrix pencils with rank at most $r$ when considered as an algebraic set.