jax.numpy.linalg.eigvals

jax.numpy.linalg.eigvals(a)[source]

Compute the eigenvalues of a general matrix.

LAX-backend implementation of eigvals(). Original docstring below.

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 – The eigenvalues, each repeated according to its multiplicity. They are not necessarily ordered, nor are they necessarily real for real matrices.

Return type

(.., M,) ndarray

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.

Notes

New in version 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