null_space — SciPy v1.15.3 Manual (original) (raw)

scipy.linalg.

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 thanrcond * 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]])