numpy.linalg.eigh

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

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

w(…, M) ndarray

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

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

The column v[:, i] is the normalized eigenvector corresponding to the eigenvalue w[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 v of (column) eigenvectors is unitary and a, w, and v satisfy the equations dot(a, v[:, i]) = w[i] * v[:, 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]])
>>> w, v = LA.eigh(a)
>>> w; v
array([0.17157288, 5.82842712])
array([[-0.92387953+0.j        , -0.38268343+0.j        ], # may vary
       [ 0.        +0.38268343j,  0.        -0.92387953j]])

>>> np.dot(a, v[:, 0]) - w[0] * v[:, 0] # verify 1st e-val/vec pair
array([5.55111512e-17+0.0000000e+00j, 0.00000000e+00+1.2490009e-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        ], # 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       ]])