numpy.linalg.eigh — NumPy v1.11 Manual (original) (raw)
numpy.linalg.eigh(a, UPLO='L')[source]¶
Return the eigenvalues and eigenvectors of a Hermitian or 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/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’). |
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| Returns: | 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]. Will return a matrix object if a is a matrix object. |
| Raises: | LinAlgError If the eigenvalue computation does not converge. |
See also
eigenvalues of symmetric or Hermitian arrays.
eigenvalues and right eigenvectors for non-symmetric arrays.
eigenvalues of non-symmetric arrays.
Notes
New in version 1.8.0.
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. [R38] The array v of (column) eigenvectors is unitary and a, w, and v satisfy the equationsdot(a, v[:, i]) = w[i] * v[:, i].
References
| [R38] | (1, 2) G. Strang, Linear Algebra and Its Applications, 2nd Ed., Orlando, FL, Academic Press, Inc., 1980, pg. 222. |
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Examples
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]]) w, v = LA.eigh(a) w; v array([ 0.17157288, 5.82842712]) array([[-0.92387953+0.j , -0.38268343+0.j ], [ 0.00000000+0.38268343j, 0.00000000-0.92387953j]])
np.dot(a, v[:, 0]) - w[0] * v[:, 0] # verify 1st e-val/vec pair array([2.77555756e-17 + 0.j, 0. + 1.38777878e-16j]) np.dot(a, v[:, 1]) - w[1] * v[:, 1] # verify 2nd e-val/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]]) w, v = LA.eigh(A) w; v array([ 0.17157288, 5.82842712]) matrix([[-0.92387953+0.j , -0.38268343+0.j ], [ 0.00000000+0.38268343j, 0.00000000-0.92387953j]])