numpy.linalg.eigh¶
-
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’). 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, (…, M, M) matrix}
The column
v[:, i]
is the normalized eigenvector corresponding to the eigenvaluew[i]
. Will return a matrix object if a is a matrix object.Raises: LinAlgError
If the eigenvalue computation does not converge.
See also
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. [R43] The array v of (column) eigenvectors is unitary and a, w, and v satisfy the equations
dot(a, v[:, i]) = w[i] * v[:, i]
.References
[R43] (1, 2) G. Strang, Linear Algebra and Its Applications, 2nd Ed., Orlando, FL, Academic Press, Inc., 1980, pg. 222. 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]])
>>> # 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; wb array([ 1., 6.]) array([ 6.+0.j, 1.+0.j]) >>> va; vb array([[-0.44721360-0.j , -0.89442719+0.j ], [ 0.00000000+0.89442719j, 0.00000000-0.4472136j ]]) array([[ 0.89442719+0.j , 0.00000000-0.4472136j], [ 0.00000000-0.4472136j, 0.89442719+0.j ]])