Module « numpy.linalg »

Fonction eigh - module numpy.linalg

Signature de la fonction eigh

def eigh(a, UPLO='L') 

Description

eigh.__doc__

    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
    -------
    w : (..., M) ndarray
        The eigenvalues in ascending order, each repeated according to
        its multiplicity.
    v : {(..., M, M) ndarray, (..., M, M) matrix}
        The column ``v[:, i]`` is the normalized eigenvector corresponding
        to the eigenvalue ``w[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
    -----

    .. versionadded:: 1.8.0

    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 `v` of (column) eigenvectors is unitary
    and `a`, `w`, and `v` satisfy the equations
    ``dot(a, v[:, i]) = w[i] * v[:, i]``.

    References
    ----------
    .. [1] G. Strang, *Linear Algebra and Its Applications*, 2nd Ed., Orlando,
           FL, Academic Press, Inc., 1980, pg. 222.

    Examples
    --------
    >>> from numpy import linalg as LA
    >>> a = np.array([[1, -2j], [2j, 5]])
    >>> a
    array([[ 1.+0.j, -0.-2.j],
           [ 0.+2.j,  5.+0.j]])
    >>> w, v = LA.eigh(a)
    >>> w; v
    array([0.17157288, 5.82842712])
    array([[-0.92387953+0.j        , -0.38268343+0.j        ], # may vary
           [ 0.        +0.38268343j,  0.        -0.92387953j]])

    >>> np.dot(a, v[:, 0]) - w[0] * v[:, 0] # verify 1st e-val/vec pair
    array([5.55111512e-17+0.0000000e+00j, 0.00000000e+00+1.2490009e-16j])
    >>> np.dot(a, v[:, 1]) - w[1] * v[:, 1] # verify 2nd e-val/vec pair
    array([0.+0.j, 0.+0.j])

    >>> A = np.matrix(a) # what happens if input is a matrix object
    >>> A
    matrix([[ 1.+0.j, -0.-2.j],
            [ 0.+2.j,  5.+0.j]])
    >>> w, v = LA.eigh(A)
    >>> w; v
    array([0.17157288, 5.82842712])
    matrix([[-0.92387953+0.j        , -0.38268343+0.j        ], # may vary
            [ 0.        +0.38268343j,  0.        -0.92387953j]])

    >>> # demonstrate the treatment of the imaginary part of the diagonal
    >>> a = np.array([[5+2j, 9-2j], [0+2j, 2-1j]])
    >>> a
    array([[5.+2.j, 9.-2.j],
           [0.+2.j, 2.-1.j]])
    >>> # with UPLO='L' this is numerically equivalent to using LA.eig() with:
    >>> b = np.array([[5.+0.j, 0.-2.j], [0.+2.j, 2.-0.j]])
    >>> b
    array([[5.+0.j, 0.-2.j],
           [0.+2.j, 2.+0.j]])
    >>> wa, va = LA.eigh(a)
    >>> wb, vb = LA.eig(b)
    >>> wa; wb
    array([1., 6.])
    array([6.+0.j, 1.+0.j])
    >>> va; vb
    array([[-0.4472136 +0.j        , -0.89442719+0.j        ], # may vary
           [ 0.        +0.89442719j,  0.        -0.4472136j ]])
    array([[ 0.89442719+0.j       , -0.        +0.4472136j],
           [-0.        +0.4472136j,  0.89442719+0.j       ]])