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# Rank-Revealing QR Factorizations and the Singular Value Decomposition

Y. P. Hong and C.-T. Pan
Mathematics of Computation
Vol. 58, No. 197 (Jan., 1992), pp. 213-232
DOI: 10.2307/2153029
Stable URL: http://www.jstor.org/stable/2153029
Page Count: 20
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## Abstract

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.

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