numpy.linalg.eigh

linalg.eigh(a, UPLO='L')[source]

Return the eigenvalues and eigenvectors of a complex Hermitian (conjugate symmetric) or a real 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(…, M, M) array

Hermitian or real symmetric matrices whose eigenvalues and eigenvectors are to be computed.

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’). Irrespective of this value only the real parts of the diagonal will be considered in the computation to preserve the notion of a Hermitian matrix. It therefore follows that the imaginary part of the diagonal will always be treated as zero.

Returns:

A namedtuple with the following attributes:

eigenvalues(…, M) ndarray

The eigenvalues in ascending order, each repeated according to its multiplicity.

eigenvectors{(…, M, M) ndarray, (…, M, M) matrix}

The column eigenvectors[:, i] is the normalized eigenvector corresponding to the eigenvalue eigenvalues[i]. Will return a matrix object if a is a matrix object.

Raises:

LinAlgError

If the eigenvalue computation does not converge.

See also

eigvalsh

eigenvalues of real symmetric or complex Hermitian (conjugate symmetric) arrays.

eig

eigenvalues and right eigenvectors for non-symmetric arrays.

eigvals

eigenvalues of non-symmetric arrays.

scipy.linalg.eigh

Similar function in SciPy (but also solves the generalized eigenvalue problem).

Notes

New in version 1.8.0.

Broadcasting rules apply, see the numpy.linalg documentation for details.

The eigenvalues/eigenvectors are computed using LAPACK routines _syevd, _heevd.

The eigenvalues of real symmetric or complex Hermitian matrices are always real. [1] The array eigenvalues of (column) eigenvectors is unitary and a, eigenvalues, and eigenvectors satisfy the equations dot(a, eigenvectors[:, i]) = eigenvalues[i] * eigenvectors[:, i].

References

[1]

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]])
>>> eigenvalues, eigenvectors = LA.eigh(a)
>>> eigenvalues
array([0.17157288, 5.82842712])
>>> eigenvectors
array([[-0.92387953+0.j        , -0.38268343+0.j        ], # may vary
       [ 0.        +0.38268343j,  0.        -0.92387953j]])

>>> (np.dot(a, eigenvectors[:, 0]) -
... eigenvalues[0] * eigenvectors[:, 0])  # verify 1st eigenval/vec pair
array([5.55111512e-17+0.0000000e+00j, 0.00000000e+00+1.2490009e-16j])
>>> (np.dot(a, eigenvectors[:, 1]) -
... eigenvalues[1] * eigenvectors[:, 1])  # verify 2nd eigenval/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]])
>>> eigenvalues, eigenvectors = LA.eigh(A)
>>> eigenvalues
array([0.17157288, 5.82842712])
>>> eigenvectors
matrix([[-0.92387953+0.j        , -0.38268343+0.j        ], # may vary
        [ 0.        +0.38268343j,  0.        -0.92387953j]])

>>> # demonstrate the treatment of the imaginary part of the diagonal
>>> a = np.array([[5+2j, 9-2j], [0+2j, 2-1j]])
>>> a
array([[5.+2.j, 9.-2.j],
       [0.+2.j, 2.-1.j]])
>>> # with UPLO='L' this is numerically equivalent to using LA.eig() with:
>>> b = np.array([[5.+0.j, 0.-2.j], [0.+2.j, 2.-0.j]])
>>> b
array([[5.+0.j, 0.-2.j],
       [0.+2.j, 2.+0.j]])
>>> wa, va = LA.eigh(a)
>>> wb, vb = LA.eig(b)
>>> wa; wb
array([1., 6.])
array([6.+0.j, 1.+0.j])
>>> va; vb
array([[-0.4472136 +0.j        , -0.89442719+0.j        ], # may vary
       [ 0.        +0.89442719j,  0.        -0.4472136j ]])
array([[ 0.89442719+0.j       , -0.        +0.4472136j],
       [-0.        +0.4472136j,  0.89442719+0.j       ]])