nlcpy.linalg.eigvals

nlcpy.linalg.eigvals(a)[source]

Computes the eigenvalues of a general matrix.

Main difference from 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.

See also

eig: Computes the eigenvalues and right eigenvectors of a square array.
eigh: Computes the eigenvalues and eigenvectors of a complex Hermitian or a real symmetric matrix.
eigvalsh: Computes the eigenvalues of a complex Hermitian or real symmetric matrix.

Note

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 nlcpy as vp
>>> x = vp.random.random()
>>> Q = vp.array([[vp.cos(x), -vp.sin(x)], [vp.sin(x), vp.cos(x)]])
>>> vp.linalg.norm(Q[0, :]), vp.linalg.norm(Q[1, :]), vp.dot(Q[0, :],Q[1, :])
(array(1.), array(1.), array(0.))

Now multiply a diagonal matrix by Q on one side and by Q.T on the other:

>>> D = vp.diag((-1,1))
>>> vp.linalg.eigvals(D)
array([-1.,  1.])
>>> A = vp.dot(Q, D)
>>> A = vp.dot(A, Q.T)
>>> vp.linalg.eigvals(A) 
array([ 1., -1.]) # random