Function cholesky

numpy/linalg/_linalg.py:790–888

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``

(a, /, *, upper=False)

@array_function_dispatch(_cholesky_dispatcher)
def 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

    .. 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},