numpy.linalg.eigvals — NumPy v1.11 Manual (original) (raw)
numpy.linalg.eigvals(a)[source]¶
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
eigenvalues and right eigenvectors of general arrays
eigenvalues of symmetric or Hermitian arrays.
eigenvalues and eigenvectors of symmetric/Hermitian 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 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 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.])