numpy.linalg.eigvalsh

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

Compute the eigenvalues of a Hermitian or real symmetric matrix.

Main difference from eigh: the eigenvectors are not computed.

Parameters:

a : (..., M, M) array_like

A complex- or real-valued matrix whose eigenvalues 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.

Raises:

LinAlgError

If the eigenvalue computation does not converge.

See also

eigh

eigenvalues and eigenvectors of symmetric/Hermitian arrays.

eigvals

eigenvalues of general real or complex arrays.

eig

eigenvalues and right eigenvectors of general real or complex arrays.

Notes

New in version 1.8.0.

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

The eigenvalues are computed using LAPACK routines _syevd, _heevd

Examples

>>> from numpy import linalg as LA
>>> a = np.array([[1, -2j], [2j, 5]])
>>> LA.eigvalsh(a)
array([ 0.17157288,  5.82842712])

>>> # 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.eigvals()
>>> # 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 = LA.eigvalsh(a)
>>> wb = LA.eigvals(b)
>>> wa; wb
array([ 1.,  6.])
array([ 6.+0.j,  1.+0.j])