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# numpy.linalg.cholesky

linalg.cholesky(a, /, *, upper=False)

  Cholesky decomposition.

  Return the lower or upper Cholesky decomposition, `L * L.H` or `U.H * U`, of the square matrix `a`, where `L` is lower-triangular, `U` is upper-triangular, and `.H` is the conjugate transpose operator (which is the ordinary transpose if `a` is real-valued). `a` must be Hermitian (symmetric if real-valued) and positive-definite. No checking is performed to verify whether `a` is Hermitian or not. In addition, only the lower or upper-triangular and diagonal elements of `a` are used. Only `L` or `U` is actually returned.

  Parameters:
    a(…, M, M) array_like

      Hermitian (symmetric if all elements are real), positive-definite input matrix.

    upper bool

      If `True`, the result must be the upper-triangular Cholesky factor. If `False`, the result must be the lower-triangular Cholesky factor. Default: `False`.

  Returns:
    L(…, M, M) array_like

      Lower or upper-triangular Cholesky factor of a. Returns a matrix object if a is a matrix object.

  Raises:
    LinAlgError

      If the decomposition fails, for example, if a is not positive-definite.

  See also

  `scipy.linalg.cholesky`

    Similar function in SciPy.

  `scipy.linalg.cholesky_banded`

    Cholesky decompose a banded Hermitian positive-definite matrix.

  `scipy.linalg.cho_factor`

    Cholesky decomposition of a matrix, to use in scipy.linalg.cho_solve.

  Notes

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

  The Cholesky decomposition is often used as a fast way of solving

  \[A \mathbf{x} = \mathbf{b}\]

  (when A is both Hermitian/symmetric and positive-definite).

  First, we solve for \(\mathbf{y}\) in

  \[L \mathbf{y} = \mathbf{b},\]

  and then for \(\mathbf{x}\) in

  \[L^{H} \mathbf{x} = \mathbf{y}.\]

  Examples

      >>> import numpy as np
      >>> A = np.array([[1,-2j],[2j,5]])
      >>> A
      array([[ 1.+0.j, -0.-2.j],
             [ 0.+2.j,  5.+0.j]])
      >>> L = np.linalg.cholesky(A)
      >>> L
      array([[1.+0.j, 0.+0.j],
             [0.+2.j, 1.+0.j]])
      >>> np.dot(L, L.T.conj()) # verify that L * L.H = A
      array([[1.+0.j, 0.-2.j],
             [0.+2.j, 5.+0.j]])
      >>> A = [[1,-2j],[2j,5]] # what happens if A is only array_like?
      >>> np.linalg.cholesky(A) # an ndarray object is returned
      array([[1.+0.j, 0.+0.j],
             [0.+2.j, 1.+0.j]])
      >>> # But a matrix object is returned if A is a matrix object
      >>> np.linalg.cholesky(np.matrix(A))
      matrix([[ 1.+0.j,  0.+0.j],
              [ 0.+2.j,  1.+0.j]])
      >>> # The upper-triangular Cholesky factor can also be obtained.
      >>> np.linalg.cholesky(A, upper=True)
      array([[1.-0.j, 0.-2.j],
             [0.-0.j, 1.-0.j]])