numpy.linalg.eigvals¶

numpy.linalg.
eigvals
(a)[source]¶ Compute the eigenvalues of a general matrix.
Main difference between
eigvals
andeig
: the eigenvectors aren’t returned. Parameters
 a(…, M, M) array_like
A complex or realvalued 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
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 asA
:>>> 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 byQ.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