nlcpy.linalg.eigvalsh

nlcpy.linalg.eigvalsh(a, UPLO='L')[source]

Computes the eigenvalues of a complex 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.

See also

eig: Computes the eigenvalues and right eigenvectors of a square array.
eigvals: Computes the eigenvalues of a general matrix.
eigh: Computes the eigenvalues and eigenvectors of a complex Hermitian or a real symmetric matrix.

Note

The eigenvalues are computed using LAPACK routines _syevd, _heevd.

Examples

>>> import nlcpy as vp
>>> a = vp.array([[1, -2j], [2j, 5]])
>>> vp.linalg.eigvalsh(a)   
array([0.17157288, 5.82842712])  # may vary

>>> # demonstrate the treatment of the imaginary part of the diagonal
>>> a = vp.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 vp.linalg.eigvals()
>>> # with:
>>> b = vp.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 = vp.linalg.eigvalsh(a)
>>> wb = vp.linalg.eigvals(b)
>>> wa; wb   
array([1., 6.])
array([6.+0.j, 1.+0.j])