Module « numpy.dual »

Fonction eig - module numpy.dual

Signature de la fonction eig

def eig(a) 

Description

eig.__doc__

    Compute the eigenvalues and right eigenvectors of a square array.

    Parameters
    ----------
    a : (..., M, M) array
        Matrices for which the eigenvalues and right eigenvectors will
        be computed

    Returns
    -------
    w : (..., M) array
        The eigenvalues, each repeated according to its multiplicity.
        The eigenvalues are not necessarily ordered. The resulting
        array will be of complex type, unless the imaginary part is
        zero in which case it will be cast to a real type. When `a`
        is real the resulting eigenvalues will be real (0 imaginary
        part) or occur in conjugate pairs

    v : (..., M, M) array
        The normalized (unit "length") eigenvectors, such that the
        column ``v[:,i]`` is the eigenvector corresponding to the
        eigenvalue ``w[i]``.

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

    See Also
    --------
    eigvals : eigenvalues of a non-symmetric array.
    eigh : eigenvalues and eigenvectors of a real symmetric or complex
           Hermitian (conjugate symmetric) array.
    eigvalsh : eigenvalues of a real symmetric or complex Hermitian
               (conjugate symmetric) array.
    scipy.linalg.eig : Similar function in SciPy that also solves the
                       generalized eigenvalue problem.
    scipy.linalg.schur : Best choice for unitary and other non-Hermitian
                         normal matrices.

    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.

    The number `w` is an eigenvalue of `a` if there exists a vector
    `v` such that ``a @ v = w * v``. Thus, the arrays `a`, `w`, and
    `v` satisfy the equations ``a @ v[:,i] = w[i] * v[:,i]``
    for :math:`i \in \{0,...,M-1\}`.

    The array `v` of eigenvectors may not be of maximum rank, that is, some
    of the columns may be linearly dependent, although round-off error may
    obscure that fact. If the eigenvalues are all different, then theoretically
    the eigenvectors are linearly independent and `a` can be diagonalized by
    a similarity transformation using `v`, i.e, ``inv(v) @ a @ v`` is diagonal.

    For non-Hermitian normal matrices the SciPy function `scipy.linalg.schur`
    is preferred because the matrix `v` is guaranteed to be unitary, which is
    not the case when using `eig`. The Schur factorization produces an
    upper triangular matrix rather than a diagonal matrix, but for normal
    matrices only the diagonal of the upper triangular matrix is needed, the
    rest is roundoff error.

    Finally, it is emphasized that `v` consists of the *right* (as in
    right-hand side) eigenvectors of `a`.  A vector `y` satisfying
    ``y.T @ a = z * y.T`` for some number `z` is called a *left*
    eigenvector of `a`, and, in general, the left and right eigenvectors
    of a matrix are not necessarily the (perhaps conjugate) transposes
    of each other.

    References
    ----------
    G. Strang, *Linear Algebra and Its Applications*, 2nd Ed., Orlando, FL,
    Academic Press, Inc., 1980, Various pp.

    Examples
    --------
    >>> from numpy import linalg as LA

    (Almost) trivial example with real e-values and e-vectors.

    >>> w, v = LA.eig(np.diag((1, 2, 3)))
    >>> w; v
    array([1., 2., 3.])
    array([[1., 0., 0.],
           [0., 1., 0.],
           [0., 0., 1.]])

    Real matrix possessing complex e-values and e-vectors; note that the
    e-values are complex conjugates of each other.

    >>> w, v = LA.eig(np.array([[1, -1], [1, 1]]))
    >>> w; v
    array([1.+1.j, 1.-1.j])
    array([[0.70710678+0.j        , 0.70710678-0.j        ],
           [0.        -0.70710678j, 0.        +0.70710678j]])

    Complex-valued matrix with real e-values (but complex-valued e-vectors);
    note that ``a.conj().T == a``, i.e., `a` is Hermitian.

    >>> a = np.array([[1, 1j], [-1j, 1]])
    >>> w, v = LA.eig(a)
    >>> w; v
    array([2.+0.j, 0.+0.j])
    array([[ 0.        +0.70710678j,  0.70710678+0.j        ], # may vary
           [ 0.70710678+0.j        , -0.        +0.70710678j]])

    Be careful about round-off error!

    >>> a = np.array([[1 + 1e-9, 0], [0, 1 - 1e-9]])
    >>> # Theor. e-values are 1 +/- 1e-9
    >>> w, v = LA.eig(a)
    >>> w; v
    array([1., 1.])
    array([[1., 0.],
           [0., 1.]])