Rank structures preserved by the QR-algorithm: the singular case

In an earlier paper we introduced the classes of polynomial and rank structures, both of them preserved by applying a (shifted) QRstep on a matrix A. In the present paper we will further investigate the case of rank structures. We will show that even if A is a singular matrix, a new QR-iterate can be constructed having the same rank structure as the matrix A itself. To this end we will introduce the concepts of effectively eliminating QR-decompositions and sparse Givens patterns, both of them being concepts of independent interest.