Generic symmetric matrix pencils with bounded rank

Abstract

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