Such matrices arise in nonlinear optimization algorithms.

For a rank deficient least squares problem, , the solution vector is not unique.

Note that the routines here compute the “thin” version of the SVD with as -by- orthogonal matrix.

This allows in-place computation and is the most commonly-used form in practice.

The presence of a zero singular value indicates that the matrix is singular.

The number of non-zero singular values indicates the rank of the matrix.

Another use of the decomposition is to compute an orthonormal basis for a set of vectors.

The first columns of form an orthonormal basis for the range of , , when has full column rank. The matrix is related to these components by, where and is the Householder vector . The algorithm used to perform the decomposition is Householder QR (Golub & Van Loan, “Matrix Computations”, Algorithm 5.2.1).

For a rank-deficient matrix, the null space of is given by the columns of corresponding to the zero singular values.