linalg.eigh

Description

Return the eigenvalues and eigenvectors of a complex Hermitian (conjugate symmetric) or a real symmetric matrix.

It returns two objects, a 1D array containing the eigenvalues of a, and a 2D square array or matrix (depending on the input type) of the corresponding eigenvectors (in columns).

Mandatory Input Parameters

Parameter	Type	Description
a	(…, M, M) array	Hermitian or real symmetric matrices

Optional Input Parameters

Parameter	Type	Default Value	Description
UPLO	{'L', 'U'},	'L'	Specifies whether the calculation is done with the lower triangular part ('L') or the upper triangular part ('U') of a. Irrespective of this value, only the real parts of the diagonal will be considered in the calculation to preserve the notion of a Hermitian matrix. It therefore follows that the imaginary part of the diagonal will always be treated as zero.

Return Value

Parameter	Type	Description
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]. A matrix object is returned if a is a matrix object.

Examples

>>> import numpy as np
>>> # Vector input
>>> a = np.array([[1,-2j], [2j, 5]])
>>> a
array([[ 1.+0.j, -0.-2.j],
       [ 0.+2.j,  5.+0.j]])
>>> w, v = np.linalg.eigh(a)
>>> w, v
(array([0.17157288, 5.82842712]), array([[-0.92387953-0.j        , -0.38268343+0.j        ],
       [ 0.        +0.38268343j,  0.        -0.92387953j]]))
>>> 
>>> np.dot(a, v[:, 0]) - w[0] * v[:, 0]
array([5.55111512e-17+0.0000000e+00j, 0.00000000e+00+1.2490009e-16j])
>>> np.dot(a, v[:, 1]) - w[1] * v[:, 1]
array([0.+0.j, 0.+0.j])
>>> 
>>> # Matrix input
>>> A = np.matrix(a)
>>> A
matrix([[ 1.+0.j, -0.-2.j],
        [ 0.+2.j,  5.+0.j]])
>>> w, v = np.linalg.eigh(A)
>>> w; v
array([0.17157288, 5.82842712])
matrix([[-0.92387953-0.j        , -0.38268343+0.j        ],
        [ 0.        +0.38268343j,  0.        -0.92387953j]])
>>> 
>>> # 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]])
>>> 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]])
>>> 
>>> np.linalg.eigh(a)
(array([1., 6.]), array([[-0.4472136 -0.j        , -0.89442719+0.j        ],
       [ 0.        +0.89442719j,  0.        -0.4472136j ]]))
>>> 
>>> np.linalg.eigh(b)
(array([1., 6.]), array([[-0.4472136 -0.j        , -0.89442719+0.j        ],
       [ 0.        +0.89442719j,  0.        -0.4472136j ]]))
>>>

Parent topic: Linear Algebra Functions