Compute the eigenvalues of a general matrix.
Main difference between eigvals and eig: the eigenvectors aren’t returned.
a ((..., M, M) array_like) – A complex- or real-valued matrix whose eigenvalues will be computed.
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
LinAlgError – If the eigenvalue computation does not converge.
New in version 1.8.0(numpy.linalg).
Broadcasting rules apply, see the numpy.linalg documentation for details.
This is implemented using the
_geevLAPACK routines which compute the eigenvalues and eigenvectors of general square arrays.
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.Thas the same eigenvalues as
>>> 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
Qon one side and by
Q.Ton 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
This method has not been implemented yet. Xorbits will try to execute it with numpy.linalg.
This docstring was copied from numpy.linalg.