The QR factorization with column pivoting does not enable us to compute
a minimum norm solution to a rank-deficient linear least squares problem
unless 
. However,
by applying further orthogonal (or unitary) transformations
from the right to the upper trapezoidal matrix
,
using the routine PxTZRZF, 
can be eliminated:
This gives the
complete orthogonal
factorization
from which the minimum norm solution can be obtained as
The matrix Z is not
formed explicitly but is represented as a product of elementary
reflectors,
as described in section 3.4.
Users need not be aware of the details of this representation,
because associated routines are provided to work with Z:
PxORMRZ (or
PxUNMRZ ) can pre- or post-multiply
a given matrix by Z or 
(
if complex).