Rank-revealing ππ factorizations and the singular value decomposition
Yoopyo Hong, C.-T. Pan Β· Mathematics of Computation Β· 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 Γ n m \times n with numerical rank r . The bounds derived in this paper that guarantee the existence of RRQR are all of order n r \sqrt {nr} , in comparison with Chanβs O ( 2 n β r ) O({2^{n - r}}) . It has been known for some time that if A is only numerically rank-one deficient, then the column permutation Ξ \Pi of A that guarantees a small r n n {r_{nn}} in the QR factorization of A Ξ A\Pi 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.