Function cholesky

numpy/linalg/linalg.py:689–780

Cholesky decomposition. Return the Cholesky decomposition, `L * L.H`, of the square matrix `a`, where `L` is lower-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) an

(a)

@array_function_dispatch(_unary_dispatcher)
def cholesky(a):
    """
    Cholesky decomposition.

    Return the Cholesky decomposition, `L * L.H`, of the square matrix `a`,
    where `L` is lower-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-triangular and diagonal elements of `a`
    are used. Only `L` is actually returned.

    Parameters
    ----------
    a : (..., M, M) array_like
        Hermitian (symmetric if all elements are real), positive-definite
        input matrix.

    Returns
    -------
    L : (..., M, M) array_like
        Lower-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
    -----

    .. versionadded:: 1.8.0

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

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

    .. math:: A \\mathbf{x} = \\mathbf{b}

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

    First, we solve for :math:`\\mathbf{y}` in

    .. math:: L \\mathbf{y} = \\mathbf{b},

    and then for :math:`\\mathbf{x}` in