Generic symmetric matrix pencils with bounded rank

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 the irreducible components of the set of $n\times n$ symmetric matrix pencils with rank at most $r$, when considered as an algebraic set, are among these closures.