scipy.linalg.

null_space#

scipy.linalg.null_space(A, rcond=None, *, overwrite_a=False, check_finite=True, lapack_driver='gesdd')[source]#

Construct an orthonormal basis for the null space of A using SVD

Parameters:

A(M, N) array_like: Input array
rcondfloat, optional: Relative condition number. Singular values s smaller than rcond * max(s) are considered zero. Default: floating point eps * max(M,N).
overwrite_abool, optional: Whether to overwrite a; may improve performance. Default is False.
check_finitebool, optional: Whether to check that the input matrix contains only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs.
lapack_driver{‘gesdd’, ‘gesvd’}, optional: Whether to use the more efficient divide-and-conquer approach ('gesdd') or general rectangular approach ('gesvd') to compute the SVD. MATLAB and Octave use the 'gesvd' approach. Default is 'gesdd'.

Returns:

Z(N, K) ndarray: Orthonormal basis for the null space of A. K = dimension of effective null space, as determined by rcond

See also

svd: Singular value decomposition of a matrix
orth: Matrix range

Examples

1-D null space:

>>> import numpy as np
>>> from scipy.linalg import null_space
>>> A = np.array([[1, 1], [1, 1]])
>>> ns = null_space(A)
>>> ns * np.copysign(1, ns[0,0])  # Remove the sign ambiguity of the vector
array([[ 0.70710678],
       [-0.70710678]])

2-D null space:

>>> from numpy.random import default_rng
>>> rng = default_rng()
>>> B = rng.random((3, 5))
>>> Z = null_space(B)
>>> Z.shape
(5, 2)
>>> np.allclose(B.dot(Z), 0)
True

The basis vectors are orthonormal (up to rounding error):

>>> Z.T.dot(Z)
array([[  1.00000000e+00,   6.92087741e-17],
       [  6.92087741e-17,   1.00000000e+00]])