numpy.linalg.eig#

linalg.eig(a)[source]#

Compute the eigenvalues and right eigenvectors of a square array.

Parameters:

a(…, M, M) array: Matrices for which the eigenvalues and right eigenvectors will be computed

Returns:

A namedtuple with the following attributes:
eigenvalues(…, M) array: The eigenvalues, each repeated according to its multiplicity. The eigenvalues are not necessarily ordered. The resulting array will be of complex type, unless the imaginary part is zero in which case it will be cast to a real type. When a is real the resulting eigenvalues will be real (0 imaginary part) or occur in conjugate pairs
eigenvectors(…, M, M) array: The normalized (unit “length”) eigenvectors, such that the column eigenvectors[:,i] is the eigenvector corresponding to the eigenvalue eigenvalues[i].

Raises:

LinAlgError: If the eigenvalue computation does not converge.

See also

eigvals: eigenvalues of a non-symmetric array.
eigh: eigenvalues and eigenvectors of a real symmetric or complex Hermitian (conjugate symmetric) array.
eigvalsh: eigenvalues of a real symmetric or complex Hermitian (conjugate symmetric) array.
scipy.linalg.eig: Similar function in SciPy that also solves the generalized eigenvalue problem.
scipy.linalg.schur: Best choice for unitary and other non-Hermitian normal matrices.

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.

The number w is an eigenvalue of a if there exists a vector v such that a @ v = w * v. Thus, the arrays a, eigenvalues, and eigenvectors satisfy the equations a @ eigenvectors[:,i] = eigenvalues[i] * eigenvectors[:,i] for \(i \in \{0,...,M-1\}\).

The array eigenvectors may not be of maximum rank, that is, some of the columns may be linearly dependent, although round-off error may obscure that fact. If the eigenvalues are all different, then theoretically the eigenvectors are linearly independent and a can be diagonalized by a similarity transformation using eigenvectors, i.e, inv(eigenvectors) @ a @ eigenvectors is diagonal.

For non-Hermitian normal matrices the SciPy function scipy.linalg.schur is preferred because the matrix eigenvectors is guaranteed to be unitary, which is not the case when using eig. The Schur factorization produces an upper triangular matrix rather than a diagonal matrix, but for normal matrices only the diagonal of the upper triangular matrix is needed, the rest is roundoff error.

Finally, it is emphasized that eigenvectors consists of the right (as in right-hand side) eigenvectors of a. A vector y satisfying y.T @ a = z * y.T for some number z is called a left eigenvector of a, and, in general, the left and right eigenvectors of a matrix are not necessarily the (perhaps conjugate) transposes of each other.

References

G. Strang, Linear Algebra and Its Applications, 2nd Ed., Orlando, FL, Academic Press, Inc., 1980, Various pp.

Examples

>>> import numpy as np
>>> from numpy import linalg as LA

(Almost) trivial example with real eigenvalues and eigenvectors.

>>> eigenvalues, eigenvectors = LA.eig(np.diag((1, 2, 3)))
>>> eigenvalues
array([1., 2., 3.])
>>> eigenvectors
array([[1., 0., 0.],
       [0., 1., 0.],
       [0., 0., 1.]])

Real matrix possessing complex eigenvalues and eigenvectors; note that the eigenvalues are complex conjugates of each other.

>>> eigenvalues, eigenvectors = LA.eig(np.array([[1, -1], [1, 1]]))
>>> eigenvalues
array([1.+1.j, 1.-1.j])
>>> eigenvectors
array([[0.70710678+0.j        , 0.70710678-0.j        ],
       [0.        -0.70710678j, 0.        +0.70710678j]])

Complex-valued matrix with real eigenvalues (but complex-valued eigenvectors); note that a.conj().T == a, i.e., a is Hermitian.

>>> a = np.array([[1, 1j], [-1j, 1]])
>>> eigenvalues, eigenvectors = LA.eig(a)
>>> eigenvalues
array([2.+0.j, 0.+0.j])
>>> eigenvectors
array([[ 0.        +0.70710678j,  0.70710678+0.j        ], # may vary
       [ 0.70710678+0.j        , -0.        +0.70710678j]])

Be careful about round-off error!

>>> a = np.array([[1 + 1e-9, 0], [0, 1 - 1e-9]])
>>> # Theor. eigenvalues are 1 +/- 1e-9
>>> eigenvalues, eigenvectors = LA.eig(a)
>>> eigenvalues
array([1., 1.])
>>> eigenvectors
array([[1., 0.],
       [0., 1.]])