Function eigh

numpy/linalg/_linalg.py:1514–1657

Return the eigenvalues and eigenvectors of a complex Hermitian (conjugate symmetric) or a real symmetric matrix. Returns two objects, a 1-D array containing the eigenvalues of `a`, and a 2-D square array or matrix (depending on the input type) of the corresponding eigenvectors

(a, UPLO='L')

@array_function_dispatch(_eigvalsh_dispatcher)
def eigh(a, UPLO='L'):
    """
    Return the eigenvalues and eigenvectors of a complex Hermitian
    (conjugate symmetric) or a real symmetric matrix.

    Returns two objects, a 1-D array containing the eigenvalues of `a`, and
    a 2-D square array or matrix (depending on the input type) of the
    corresponding eigenvectors (in columns).

    Parameters
    ----------
    a : (..., M, M) array
        Hermitian or real symmetric matrices whose eigenvalues and
        eigenvectors are to be computed.
    UPLO : {'L', 'U'}, optional
        Specifies whether the calculation is done with the lower triangular
        part of `a` ('L', default) or the upper triangular part ('U').
        Irrespective of this value only the real parts of the diagonal will
        be considered in the computation to preserve the notion of a Hermitian
        matrix. It therefore follows that the imaginary part of the diagonal
        will always be treated as zero.

    Returns
    -------
    A namedtuple with the following attributes:

    eigenvalues : (..., M) ndarray
        The eigenvalues in ascending order, each repeated according to
        its multiplicity.
    eigenvectors : {(..., M, M) ndarray, (..., M, M) matrix}
        The column ``eigenvectors[:, i]`` is the normalized eigenvector
        corresponding to the eigenvalue ``eigenvalues[i]``.  Will return a
        matrix object if `a` is a matrix object.

    Raises
    ------
    LinAlgError
        If the eigenvalue computation does not converge.

    See Also
    --------
    eigvalsh : eigenvalues of real symmetric or complex Hermitian
               (conjugate symmetric) arrays.
    eig : eigenvalues and right eigenvectors for non-symmetric arrays.
    eigvals : eigenvalues of non-symmetric arrays.
    scipy.linalg.eigh : Similar function in SciPy (but also solves the
                        generalized eigenvalue problem).

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

    The eigenvalues/eigenvectors are computed using LAPACK routines ``_syevd``,
    ``_heevd``.

    The eigenvalues of real symmetric or complex Hermitian matrices are always
    real. [1]_ The array `eigenvalues` of (column) eigenvectors is unitary and