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 : array_like, shape (M, M)

A complex- or real-valued matrix whose eigenvalues will be computed.

Returns:

w : ndarray, shape (M,)

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

This is a simple interface to the LAPACK routines dgeev and zgeev that sets those routines’ flags to return only the eigenvalues of general real and complex arrays, respectively.

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.])