Rank-Revealing QR Factorizations and the Singular Value Decomposition
Y. P. Hong, Chenjian Pan · Mathematics of Computation · 1992
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 V/ir, in comparison with Chan's 0(2n-r) .It has been known for some time that if A is only numerically rank-one deficient, then the column permutation II of A that guarantees a small rnn in the QR factorization of Afl 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.