Compute the eigenvalues of a general matrix.
- a : (…, M, M) array_like
A complex- or real-valued matrix whose eigenvalues will be computed.
- w : (…, M,) ndarray
The eigenvalues, each repeated according to its multiplicity. They are not necessarily ordered, nor are they necessarily real for real matrices.
If the eigenvalue computation does not converge.
Broadcasting rules apply, see the
numpy.linalgdocumentation for details.
This is implemented using the _geev LAPACK 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 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.])