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scipy.linalg.

# null_space

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

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

  The documentation is written assuming array arguments are of specified “core” shapes. However, array argument(s) of this function may have additional “batch” dimensions prepended to the core shape. In this case, the array is treated as a batch of lower-dimensional slices; see Batched Linear Operations for details. Note that calls with zero-size batches are unsupported and will raise a `ValueError`.

  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).

    overwrite_a bool, optional

      Whether to overwrite a; may improve performance. Default is False. See overwrite_* arguments for details.

    check_finite bool, 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]])