  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.
