scipy.linalg.eig#
- scipy.linalg.eig(a, b=None, left=False, right=True, overwrite_a=False, overwrite_b=False, check_finite=True, homogeneous_eigvals=False)[source]#
Solve an ordinary or generalized eigenvalue problem of a square matrix.
Find eigenvalues w and right or left eigenvectors of a general matrix:
a vr[:,i] = w[i] b vr[:,i] a.H vl[:,i] = w[i].conj() b.H vl[:,i]
where
.H
is the Hermitian conjugation.- Parameters:
- a(M, M) array_like
A complex or real matrix whose eigenvalues and eigenvectors will be computed.
- b(M, M) array_like, optional
Right-hand side matrix in a generalized eigenvalue problem. Default is None, identity matrix is assumed.
- leftbool, optional
Whether to calculate and return left eigenvectors. Default is False.
- rightbool, optional
Whether to calculate and return right eigenvectors. Default is True.
- overwrite_abool, optional
Whether to overwrite a; may improve performance. Default is False.
- overwrite_bbool, optional
Whether to overwrite b; may improve performance. Default is False.
- check_finitebool, optional
Whether to check that the input matrices contain only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs.
- homogeneous_eigvalsbool, optional
If True, return the eigenvalues in homogeneous coordinates. In this case
w
is a (2, M) array so that:w[1,i] a vr[:,i] = w[0,i] b vr[:,i]
Default is False.
- Returns:
- w(M,) or (2, M) double or complex ndarray
The eigenvalues, each repeated according to its multiplicity. The shape is (M,) unless
homogeneous_eigvals=True
.- vl(M, M) double or complex ndarray
The left eigenvector corresponding to the eigenvalue
w[i]
is the columnvl[:,i]
. Only returned ifleft=True
. The left eigenvector is not normalized.- vr(M, M) double or complex ndarray
The normalized right eigenvector corresponding to the eigenvalue
w[i]
is the columnvr[:,i]
. Only returned ifright=True
.
- Raises:
- LinAlgError
If eigenvalue computation does not converge.
See also
eigvals
eigenvalues of general arrays
eigh
Eigenvalues and right eigenvectors for symmetric/Hermitian arrays.
eig_banded
eigenvalues and right eigenvectors for symmetric/Hermitian band matrices
eigh_tridiagonal
eigenvalues and right eiegenvectors for symmetric/Hermitian tridiagonal matrices
Examples
>>> import numpy as np >>> from scipy import linalg >>> a = np.array([[0., -1.], [1., 0.]]) >>> linalg.eigvals(a) array([0.+1.j, 0.-1.j])
>>> b = np.array([[0., 1.], [1., 1.]]) >>> linalg.eigvals(a, b) array([ 1.+0.j, -1.+0.j])
>>> a = np.array([[3., 0., 0.], [0., 8., 0.], [0., 0., 7.]]) >>> linalg.eigvals(a, homogeneous_eigvals=True) array([[3.+0.j, 8.+0.j, 7.+0.j], [1.+0.j, 1.+0.j, 1.+0.j]])
>>> a = np.array([[0., -1.], [1., 0.]]) >>> linalg.eigvals(a) == linalg.eig(a)[0] array([ True, True]) >>> linalg.eig(a, left=True, right=False)[1] # normalized left eigenvector array([[-0.70710678+0.j , -0.70710678-0.j ], [-0. +0.70710678j, -0. -0.70710678j]]) >>> linalg.eig(a, left=False, right=True)[1] # normalized right eigenvector array([[0.70710678+0.j , 0.70710678-0.j ], [0. -0.70710678j, 0. +0.70710678j]])