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
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 Chan’s
O
(
2
n
−
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
Π
\Pi
of A that guarantees a small
r
n
n
{r_{nn}}
in the QR factorization of
A
Π
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.