numpy.linalg.eigvals

numpy.linalg.eigvals(a)

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 symmetric or Hermitian arrays.

eigh

eigenvalues and eigenvectors of symmetric/Hermitian arrays.

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:

>>> 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.])