# 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
    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 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 v of (column) eigenvectors is unitary and a, w, and v satisfy the equations `dot(a, v[:, i]) = w[i] * v[:, 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]])
      >>> w, v = LA.eigh(a)
      >>> w; v
      array([0.17157288, 5.82842712])
      array([[-0.92387953+0.j        , -0.38268343+0.j        ], # may vary
             [ 0.        +0.38268343j,  0.        -0.92387953j]])

      >>> np.dot(a, v[:, 0]) - w[0] * v[:, 0] # verify 1st e-val/vec pair
      array([5.55111512e-17+0.0000000e+00j, 0.00000000e+00+1.2490009e-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        ], # 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       ]])

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