# scipy.linalg.eigvalsh

scipy.linalg.eigvalsh(a, b=None, lower=True, overwrite_a=False, overwrite_b=False, turbo=False, eigvals=None, type=1, check_finite=True, subset_by_index=None, subset_by_value=None, driver=None)

  Solves a standard or generalized eigenvalue problem for a complex Hermitian or real symmetric matrix.

  Find eigenvalues array `w` of array `a`, where `b` is positive definite such that for every eigenvalue λ (i-th entry of w) and its eigenvector vi (i-th column of v) satisfies:

                    a @ vi = λ * b @ vi
      vi.conj().T @ a @ vi = λ
      vi.conj().T @ b @ vi = 1

  In the standard problem, b is assumed to be the identity matrix.

  Parameters:
    a(M, M) array_like

      A complex Hermitian or real symmetric matrix whose eigenvalues will be computed.

    b(M, M) array_like, optional

      A complex Hermitian or real symmetric definite positive matrix in. If omitted, identity matrix is assumed.

    lower bool, optional

      Whether the pertinent array data is taken from the lower or upper triangle of `a` and, if applicable, `b`. (Default: lower)

    overwrite_a bool, optional

      Whether to overwrite data in `a` (may improve performance). Default is False.

    overwrite_b bool, optional

      Whether to overwrite data in `b` (may improve performance). Default is False.

    type int, optional

      For the generalized problems, this keyword specifies the problem type to be solved for `w` and `v` (only takes 1, 2, 3 as possible inputs):

          1 =>     a @ v = w @ b @ v
          2 => a @ b @ v = w @ v
          3 => b @ a @ v = w @ v

      This keyword is ignored for standard problems.

    check_finite bool, optional

      Whether to check that the input matrices contain only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs.

    subset_by_index iterable, optional

      If provided, this two-element iterable defines the start and the end indices of the desired eigenvalues (ascending order and 0-indexed). To return only the second smallest to fifth smallest eigenvalues, `[1, 4]` is used. `[n-3, n-1]` returns the largest three. Only available with “evr”, “evx”, and “gvx” drivers. The entries are directly converted to integers via `int()`.

    subset_by_value iterable, optional

      If provided, this two-element iterable defines the half-open interval `(a, b]` that, if any, only the eigenvalues between these values are returned. Only available with “evr”, “evx”, and “gvx” drivers. Use `np.inf` for the unconstrained ends.

    driver str, optional

      Defines which LAPACK driver should be used. Valid options are “ev”, “evd”, “evr”, “evx” for standard problems and “gv”, “gvd”, “gvx” for generalized (where b is not None) problems. See the Notes section of `scipy.linalg.eigh`.

    turbo bool, optional, deprecated

      Deprecated since version 1.5.0: ‘eigvalsh’ keyword argument turbo is deprecated in favor of `driver=gvd` option and will be removed in SciPy 1.12.0.

    eigvals tuple (lo, hi), optional

      Deprecated since version 1.5.0: ‘eigvalsh’ keyword argument `eigvals` is deprecated in favor of subset_by_index option and will be removed in SciPy 1.12.0.

  Returns:
    w(N,) ndarray

      The `N` (`1<=N<=M`) selected eigenvalues, in ascending order, each repeated according to its multiplicity.

  Raises:
    LinAlgError

      If eigenvalue computation does not converge, an error occurred, or b matrix is not definite positive. Note that if input matrices are not symmetric or Hermitian, no error will be reported but results will be wrong.

  See also

  `eigh`

    eigenvalues and right eigenvectors for symmetric/Hermitian arrays

  `eigvals`

    eigenvalues of general arrays

  `eigvals_banded`

    eigenvalues for symmetric/Hermitian band matrices

  `eigvalsh_tridiagonal`

    eigenvalues of symmetric/Hermitian tridiagonal matrices

  Notes

  This function does not check the input array for being Hermitian/symmetric in order to allow for representing arrays with only their upper/lower triangular parts.

  This function serves as a one-liner shorthand for `scipy.linalg.eigh` with the option `eigvals_only=True` to get the eigenvalues and not the eigenvectors. Here it is kept as a legacy convenience. It might be beneficial to use the main function to have full control and to be a bit more pythonic.

  Examples

  For more examples see `scipy.linalg.eigh`.

      >>> import numpy as np
      >>> from scipy.linalg import eigvalsh
      >>> A = np.array([[6, 3, 1, 5], [3, 0, 5, 1], [1, 5, 6, 2], [5, 1, 2, 2]])
      >>> w = eigvalsh(A)
      >>> w
      array([-3.74637491, -0.76263923,  6.08502336, 12.42399079])