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 × n with numerical rank r. The bounds derived in this paper that guarantee the existence of RRQR are all of order $\sqrt{nr}$, in comparison with Chan's O(2n - r). It has been known for some time that if A is only numerically rank-one deficient, then the column permutation Π of A that guarantees a small rnn in the QR factorization of AΠ 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.