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# numpy.linalg.eigh

`numpy.linalg.``eigh`(a, UPLO='L')

  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’).
  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

  `eigvalsh`
    eigenvalues of symmetric or Hermitian arrays.
  `eig`
    eigenvalues and right eigenvectors for non-symmetric arrays.
  `eigvals`
    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 equations `dot(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.

  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]])