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The set of mxn singular matrix pencils with normal rank at most r is analgebraic set with r+1 irreducible components. These components are the closureof the orbits (under strict equivalence) of r+1 matrix pencils which are inKronecker canonical form. In this paper, we provide a new explicit descriptionof each of these irreducible components which is a parametrization of eachcomponent. Therefore one can explicitly construct any pencil in each of thesecomponents. The new description of each of these irreducible componentsconsists of the sum of r rank-1 matrix pencils, namely, a column polynomialvector 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 rowvectors with zero degree determines each irreducible component.

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