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# numpy.linalg.svd

`numpy.linalg.``svd`(a, full_matrices=1, compute_uv=1)

  Singular Value Decomposition.

  Factors the matrix a as `u * np.diag(s) * v`, where u and v are unitary and s is a 1-d array of a‘s singular values.

  Parameters: | a : (..., M, N) array_like A real or complex matrix of shape (M, N) . full_matrices : bool, optional If True (default), u and v have the shapes (M, M) and (N, N), respectively. Otherwise, the shapes are (M, K) and (K, N), respectively, where K = min(M, N). compute_uv : bool, optional Whether or not to compute u and v in addition to s. True by default.
  Returns: | u : { (..., M, M), (..., M, K) } array Unitary matrices. The actual shape depends on the value of `full_matrices`. Only returned when `compute_uv` is True. s : (..., K) array The singular values for every matrix, sorted in descending order. v : { (..., N, N), (..., K, N) } array Unitary matrices. The actual shape depends on the value of `full_matrices`. Only returned when `compute_uv` is True.
  Raises: | LinAlgError : If SVD computation does not converge.

  Notes

  Broadcasting rules apply, see the `numpy.linalg` documentation for details.

  The decomposition is performed using LAPACK routine _gesdd

  The SVD is commonly written as `a = U S V.H`. The v returned by this function is `V.H` and `u = U`.

  If `U` is a unitary matrix, it means that it satisfies `U.H = inv(U)`.

  The rows of v are the eigenvectors of `a.H a`. The columns of u are the eigenvectors of `a a.H`. For row `i` in v and column `i` in u, the corresponding eigenvalue is `s[i]**2`.

  If a is a matrix object (as opposed to an ndarray), then so are all the return values.

  Examples

      >>> a = np.random.randn(9, 6) + 1j*np.random.randn(9, 6)

  Reconstruction based on full SVD:

      >>> U, s, V = np.linalg.svd(a, full_matrices=True)
      >>> U.shape, V.shape, s.shape
      ((9, 9), (6, 6), (6,))
      >>> S = np.zeros((9, 6), dtype=complex)
      >>> S[:6, :6] = np.diag(s)
      >>> np.allclose(a, np.dot(U, np.dot(S, V)))
      True

  Reconstruction based on reduced SVD:

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