numpy.linalg.eigh

numpy.linalg.eigh(a, UPLO='L')

Return the eigenvalues and eigenvectors of a Hermitian or symmetric matrix.

Returns two objects, a 1-D array containing the eigenvalues of a, and a 2-D square array or matrix (depending on the input type) of the corresponding eigenvectors (in columns).

Parameters:

a : array_like, shape (M, M)

A complex Hermitian or real symmetric matrix.

UPLO : {‘L’, ‘U’}, optional

Specifies whether the calculation is done with the lower triangular part of a (‘L’, default) or the upper triangular part (‘U’).

Returns:

w : ndarray, shape (M,)

The eigenvalues, not necessarily ordered.

v : ndarray, or matrix object if a is, shape (M, M)

The column v[:, i] is the normalized eigenvector corresponding to the eigenvalue w[i].

Raises:

LinAlgError :

If the eigenvalue computation does not converge.

See also

eigvalsh

eigenvalues of symmetric or Hermitian arrays.

eig

eigenvalues and right eigenvectors for non-symmetric arrays.

eigvals

eigenvalues of non-symmetric arrays.

Notes

This is a simple interface to the LAPACK routines dsyevd and zheevd, which compute the eigenvalues and eigenvectors of real symmetric and complex Hermitian arrays, respectively.

The eigenvalues of real symmetric or complex Hermitian matrices are always real. [R47] The array v of (column) eigenvectors is unitary and a, w, and v satisfy the equations dot(a, v[:, i]) = w[i] * v[:, i].

References

[R47]

(1, 2) G. Strang, Linear Algebra and Its Applications, 2nd Ed., Orlando, FL, Academic Press, Inc., 1980, pg. 222.

Examples

>>> from numpy import linalg as LA
>>> a = np.array([[1, -2j], [2j, 5]])
>>> a
array([[ 1.+0.j,  0.-2.j],
       [ 0.+2.j,  5.+0.j]])
>>> w, v = LA.eigh(a)
>>> w; v
array([ 0.17157288,  5.82842712])
array([[-0.92387953+0.j        , -0.38268343+0.j        ],
       [ 0.00000000+0.38268343j,  0.00000000-0.92387953j]])

>>> np.dot(a, v[:, 0]) - w[0] * v[:, 0] # verify 1st e-val/vec pair
array([2.77555756e-17 + 0.j, 0. + 1.38777878e-16j])
>>> np.dot(a, v[:, 1]) - w[1] * v[:, 1] # verify 2nd e-val/vec pair
array([ 0.+0.j,  0.+0.j])

>>> A = np.matrix(a) # what happens if input is a matrix object
>>> A
matrix([[ 1.+0.j,  0.-2.j],
        [ 0.+2.j,  5.+0.j]])
>>> w, v = LA.eigh(A)
>>> w; v
array([ 0.17157288,  5.82842712])
matrix([[-0.92387953+0.j        , -0.38268343+0.j        ],
        [ 0.00000000+0.38268343j,  0.00000000-0.92387953j]])