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null_space(A, rcond=None)

Parameters

A : (M, N) array_like

Input array

rcond : float, optional

Relative condition number. Singular values s smaller than rcond * max(s) are considered zero. Default: floating point eps * max(M,N).

Returns

Z : (N, K) ndarray

Orthonormal basis for the null space of A. K = dimension of effective null space, as determined by rcond

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

See Also

orth

Matrix range

svd

Singular value decomposition of a matrix

Examples

1-D null space:

>>> from scipy.linalg import null_space
... A = np.array([[1, 1], [1, 1]])
... ns = null_space(A)
... ns * np.sign(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]])
See :

Back References

The following pages refer to to this document either explicitly or contain code examples using this.

scipy.linalg._decomp_svd.null_space scipy.linalg._decomp_svd.orth

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GitHub : /scipy/linalg/_decomp_svd.py#331
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