Rank-Revealing QR Factorizations and the Singular Value Decomposition

T. Chan has noted that, even when the singular value decomposition of a matrix A is known, it is still not obvious how to find a rank-revealing QR factorization (RRQR) of A if A has numerical rank deficiency. This paper offers a constructive proof of the existence of the RRQR factorization of any matrix A of size m x n with numerical rank r. The bounds derived in this paper that guarantee the existence of RRQR are all of order yfñr, in comparison with Chan's 0(2"~r). It has been known for some time that if A is only numerically rank-one deficient, then the column permutation n of A that guarantees a small rn„ in the QR factorization of All can be obtained by inspecting the size of the elements of the right singular vector of A corresponding to the smallest singular value of A . To some extent, our paper generalizes this well-known result. 0. Introduction We consider the interplay between two important matrix decompositions: the singular value decomposition and the QR factorization of a matrix A. In particular, we are interested in the case when A is singular or nearly singular. It is well known that for any A e Rmxn (a real matrix with m rows and « columns, where without loss of generality we assume m > n) there are orthogonal matrices U and V such that