An explicit description of the irreducible components of the set of matrix pencils with bounded normal rank

TL;DR

This paper provides an explicit parametrization of the irreducible components of the set of matrix pencils with bounded normal rank, enabling direct construction of any pencil within these components.

Contribution

It introduces a new explicit description of each irreducible component as sums of rank-1 pencils with specific degree constraints, improving understanding and construction.

Findings

Explicit parametrization of irreducible components

Construction method for any pencil in the components

Clarification of the structure of matrix pencils with bounded normal rank

Abstract

The set of mxn singular matrix pencils with normal rank at most r is an algebraic set with r+1 irreducible components. These components are the closure of the orbits (under strict equivalence) of r+1 matrix pencils which are in Kronecker canonical form. In this paper, we provide a new explicit description of each of these irreducible components which is a parametrization of each component. Therefore one can explicitly construct any pencil in each of these components. The new description of each of these irreducible components consists of the sum of r rank-1 matrix pencils, namely, a column polynomial vector of degree at most 1 times a row polynomial vector of degree at most 1, where we impose one of these two vectors to have degree zero. The number of row vectors with zero degree determines each irreducible component.