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Module « numpy.linalg »

Fonction eigvals - module numpy.linalg

Signature de la fonction eigvals

def eigvals(a) 

Description

help(numpy.linalg.eigvals)

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
-----
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``:

>>> import numpy as np
>>> 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

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