# numpy.linalg.eigh

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

  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

  `eigvalsh`

    eigenvalues of real symmetric or complex Hermitian (conjugate symmetric) arrays.

  `eig`

    eigenvalues and right eigenvectors for non-symmetric arrays.

  `eigvals`

    eigenvalues of non-symmetric arrays.

  `scipy.linalg.eigh`

    Similar function in SciPy (but also solves the generalized eigenvalue problem).

  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. [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

      >>> 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; wb
      array([1., 6.])
      array([6.+0.j, 1.+0.j])
      >>> va; vb
      array([[-0.4472136 +0.j        , -0.89442719+0.j        ], # may vary
             [ 0.        +0.89442719j,  0.        -0.4472136j ]])
      array([[ 0.89442719+0.j       , -0.        +0.4472136j],
             [-0.        +0.4472136j,  0.89442719+0.j       ]])