Compute the eigenvalues of a general matrix.
Main difference between eigvals and eig: the eigenvectors aren’t returned.
A complex- or real-valued matrix whose eigenvalues will be computed.
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.
See also
eig
eigenvalues and right eigenvectors of general arrays
eigvalsh
eigenvalues of real symmetric or complex Hermitian (conjugate symmetric) arrays.
eigh
eigenvalues and eigenvectors of real symmetric or complex Hermitian (conjugate symmetric) arrays.
scipy.linalg.eigvals
Similar function in SciPy.
Notes
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 as A:
>>> import numpy as np >>> 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.]) # random
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