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scipy.linalg.

# svd

scipy.linalg.svd(a, full_matrices=True, compute_uv=True, overwrite_a=False, check_finite=True, lapack_driver='gesdd')

  Singular Value Decomposition.

  Factorizes the matrix a into two unitary matrices `U` and `Vh`, and a 1-D array `s` of singular values (real, non-negative) such that `a == U @ S @ Vh`, where `S` is a suitably shaped matrix of zeros with main diagonal `s`.

  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

      Matrix to decompose.

    full_matrices bool, optional

      If True (default), U and Vh are of shape `(M, M)`, `(N, N)`. If False, the shapes are `(M, K)` and `(K, N)`, where `K = min(M, N)`.

    compute_uv bool, optional

      Whether to compute also `U` and `Vh` in addition to `s`. Default is True.

    overwrite_a bool, optional

      Whether to overwrite a; may improve performance. Default is False.

    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:
    U ndarray

      Unitary matrix having left singular vectors as columns. Of shape `(M, M)` or `(M, K)`, depending on full_matrices.

    s ndarray

      The singular values, sorted in non-increasing order. Of shape (K,), with `K = min(M, N)`.

    Vh ndarray

      Unitary matrix having right singular vectors as rows. Of shape `(N, N)` or `(K, N)` depending on full_matrices.

    For `compute_uv=False`, only `s` is returned.
  Raises:
    LinAlgError

      If SVD computation does not converge.

  See also

  `svdvals`

    Compute singular values of a matrix.

  `diagsvd`

    Construct the Sigma matrix, given the vector s.

  Examples

      >>> import numpy as np
      >>> from scipy import linalg
      >>> rng = np.random.default_rng()
      >>> m, n = 9, 6
      >>> a = rng.standard_normal((m, n)) + 1.j*rng.standard_normal((m, n))
      >>> U, s, Vh = linalg.svd(a)
      >>> U.shape,  s.shape, Vh.shape
      ((9, 9), (6,), (6, 6))

  Reconstruct the original matrix from the decomposition:

      >>> sigma = np.zeros((m, n))
      >>> for i in range(min(m, n)):
      ...     sigma[i, i] = s[i]
      >>> a1 = np.dot(U, np.dot(sigma, Vh))
      >>> np.allclose(a, a1)
      True

  Alternatively, use `full_matrices=False` (notice that the shape of `U` is then `(m, n)` instead of `(m, m)`):

      >>> U, s, Vh = linalg.svd(a, full_matrices=False)
      >>> U.shape, s.shape, Vh.shape
      ((9, 6), (6,), (6, 6))
      >>> S = np.diag(s)
      >>> np.allclose(a, np.dot(U, np.dot(S, Vh)))
      True

      >>> s2 = linalg.svd(a, compute_uv=False)
      >>> np.allclose(s, s2)
      True