Module « numpy.dual »

Fonction eigvals - module numpy.dual

Signature de la fonction eigvals

def eigvals(a) 

Description

eigvals.__doc__

    Compute the eigenvalues of a general matrix.

    Main difference between `eigvals` and `eig`: the eigenvectors aren't
    returned.

    Parameters
    ----------
    a : (..., M, M) array_like
        A complex- or real-valued matrix whose eigenvalues will be computed.

    Returns
    -------
    w : (..., M,) ndarray
        The eigenvalues, each repeated according to its multiplicity.
        They are not necessarily ordered, nor are they necessarily
        real for real matrices.

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

    See Also
    --------
    eig : eigenvalues and right eigenvectors of general arrays
    eigvalsh : eigenvalues of real symmetric or complex Hermitian
               (conjugate symmetric) arrays.
    eigh : eigenvalues and eigenvectors of real symmetric or complex
           Hermitian (conjugate symmetric) arrays.
    scipy.linalg.eigvals : Similar function in SciPy.

    Notes
    -----

    .. versionadded:: 1.8.0

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

    This is implemented using the ``_geev`` LAPACK routines which compute
    the eigenvalues and eigenvectors of general square arrays.

    Examples
    --------
    Illustration, using the fact that the eigenvalues of a diagonal matrix
    are its diagonal elements, that multiplying a matrix on the left
    by an orthogonal matrix, `Q`, and on the right by `Q.T` (the transpose
    of `Q`), preserves the eigenvalues of the "middle" matrix.  In other words,
    if `Q` is orthogonal, then ``Q * A * Q.T`` has the same eigenvalues as
    ``A``:

    >>> from numpy import linalg as LA
    >>> x = np.random.random()
    >>> Q = np.array([[np.cos(x), -np.sin(x)], [np.sin(x), np.cos(x)]])
    >>> LA.norm(Q[0, :]), LA.norm(Q[1, :]), np.dot(Q[0, :],Q[1, :])
    (1.0, 1.0, 0.0)

    Now multiply a diagonal matrix by ``Q`` on one side and by ``Q.T`` on the other:

    >>> D = np.diag((-1,1))
    >>> LA.eigvals(D)
    array([-1.,  1.])
    >>> A = np.dot(Q, D)
    >>> A = np.dot(A, Q.T)
    >>> LA.eigvals(A)
    array([ 1., -1.]) # random