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### Complete Orthogonal Factorization

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).