numpy.linalg.eigh — NumPy v2.2 Manual (original) (raw)
linalg.eigh(a, UPLO='L')[source]#
Return the eigenvalues and eigenvectors of a complex Hermitian (conjugate symmetric) 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
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:
A namedtuple with the following attributes:
eigenvalues(…, M) ndarray
The 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 eigenvalue eigenvalues[i]. Will return a matrix object if a is a matrix object.
Raises:
LinAlgError
If the eigenvalue computation does not converge.
See also
eigenvalues of real symmetric or complex Hermitian (conjugate symmetric) arrays.
eigenvalues and right eigenvectors for non-symmetric arrays.
eigenvalues of non-symmetric arrays.
Similar function in SciPy (but also solves the generalized eigenvalue problem).
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].
References
[1]
G. Strang, Linear Algebra and Its Applications, 2nd Ed., Orlando, FL, Academic Press, Inc., 1980, pg. 222.
Examples
import numpy as np from numpy import linalg as LA a = np.array([[1, -2j], [2j, 5]]) a array([[ 1.+0.j, -0.-2.j], [ 0.+2.j, 5.+0.j]]) eigenvalues, eigenvectors = LA.eigh(a) eigenvalues array([0.17157288, 5.82842712]) eigenvectors array([[-0.92387953+0.j , -0.38268343+0.j ], # may vary [ 0. +0.38268343j, 0. -0.92387953j]])
(np.dot(a, eigenvectors[:, 0]) - ... eigenvalues[0] * eigenvectors[:, 0]) # verify 1st eigenval/vec pair array([5.55111512e-17+0.0000000e+00j, 0.00000000e+00+1.2490009e-16j]) (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 matrix([[ 1.+0.j, -0.-2.j], [ 0.+2.j, 5.+0.j]]) eigenvalues, eigenvectors = LA.eigh(A) eigenvalues array([0.17157288, 5.82842712]) eigenvectors matrix([[-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 = 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.eig() 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, va = LA.eigh(a) wb, vb = LA.eig(b) wa array([1., 6.]) wb array([6.+0.j, 1.+0.j]) va array([[-0.4472136 +0.j , -0.89442719+0.j ], # may vary [ 0. +0.89442719j, 0. -0.4472136j ]]) vb array([[ 0.89442719+0.j , -0. +0.4472136j], [-0. +0.4472136j, 0.89442719+0.j ]])