bundles / numpy 2.4.3 / numpy / linalg / eigh
_ArrayFunctionDispatcher
numpy.linalg:eigh
source: /numpy/linalg/_linalg.py:1515
Signature
def eigh(a, UPLO = L)Summary
Return the eigenvalues and eigenvectors of a complex Hermitian (conjugate symmetric) or a real symmetric matrix.
Extended Summary
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) arrayHermitian or real symmetric matrices whose eigenvalues and eigenvectors are to be computed.
UPLO: {'L', 'U'}, optionalSpecifies 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
: A namedtuple with the following attributes:eigenvalues: (..., M) ndarrayThe eigenvalues in ascending order, each repeated according to its multiplicity.
eigenvectors: {(..., M, M) ndarray, (..., M, M) matrix}The column
eigenvectors[:, i]is the normalized eigenvector corresponding to the eigenvalueeigenvalues[i]. Will return a matrix object ifais a matrix object.
Raises
: LinAlgErrorIf the eigenvalue computation does not converge.
Notes
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 eigenvalues of (column) eigenvectors is unitary and a, eigenvalues, and eigenvectors satisfy the equations dot(a, eigenvectors[:, i]) = eigenvalues[i] * eigenvectors[:, i].
Examples
import numpy as np from numpy import linalg as LA a = np.array([[1, -2j], [2j, 5]]) a eigenvalues, eigenvectors = LA.eigh(a) eigenvalues✓
array([0.17157288, 5.82842712])
eigenvectors
✗array([[-0.92387953+0.j , -0.38268343+0.j ],
[ 0. +0.38268343j, 0. -0.92387953j]])
(np.dot(a, eigenvectors[:, 0]) - eigenvalues[0] * eigenvectors[:, 0]) # verify 1st eigenval/vec pair (np.dot(a, eigenvectors[:, 1]) - eigenvalues[1] * eigenvectors[:, 1]) # verify 2nd eigenval/vec pair✓
array([0.+0.j, 0.+0.j])
A = np.matrix(a) # what happens if input is a matrix object A eigenvalues, eigenvectors = LA.eigh(A) eigenvalues✓
array([0.17157288, 5.82842712])
eigenvectors
✗matrix([[-0.92387953+0.j , -0.38268343+0.j ],
[ 0. +0.38268343j, 0. -0.92387953j]])
a = np.array([[5+2j, 9-2j], [0+2j, 2-1j]]) a 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([6.+0.j, 1.+0.j])
va
✗array([[-0.4472136 +0.j , -0.89442719+0.j ],
[ 0. +0.89442719j, 0. -0.4472136j ]])
vb
✓array([[ 0.89442719+0.j , -0. +0.4472136j],
[-0. +0.4472136j, 0.89442719+0.j ]])
See also
Aliases
numpy.linalg.eigh