nlcpy.linalg.eigh

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

Computes the eigenvalues and eigenvectors of a complex Hermitian 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_like

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

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

参考

eigvalsh

Computes the eigenvalues of a complex Hermitian or real symmetric matrix.

eig

Computes the eigenvalues and right eigenvectors of a square array.

eigvals

Computes the eigenvalues of a general matrix.

注釈

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

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

Examples

>>> import nlcpy as vp
>>> a = vp.array([[1, -2j], [2j, 5]])
>>> a
array([[ 1.+0.j, -0.-2.j],
       [ 0.+2.j,  5.+0.j]])
>>> w, v = vp.linalg.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]])

>>> # 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 LA.eig() 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, va = vp.linalg.eigh(a)
>>> wb, vb = vp.linalg.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        ],
       [ 0.        +0.89442719j,  0.        -0.4472136j ]])
array([[ 0.89442719+0.j       ,  0.        +0.4472136j],
       [-0.        +0.4472136j,  0.89442719+0.j       ]])