# scipy.linalg.eig_banded

scipy.linalg.eig_banded(a_band, lower=False, eigvals_only=False, overwrite_a_band=False, select='a', select_range=None, max_ev=0, check_finite=True)

  Solve real symmetric or complex Hermitian band matrix eigenvalue problem.

  Find eigenvalues w and optionally right eigenvectors v of a:

      a v[:,i] = w[i] v[:,i]
      v.H v    = identity

  The matrix a is stored in a_band either in lower diagonal or upper diagonal ordered form:

  > a_band[u + i - j, j] == a[i,j] (if upper form; i <= j) a_band[ i - j, j] == a[i,j] (if lower form; i >= j)

  where u is the number of bands above the diagonal.

  Example of a_band (shape of a is (6,6), u=2):

      upper form:
      *   *   a02 a13 a24 a35
      *   a01 a12 a23 a34 a45
      a00 a11 a22 a33 a44 a55

      lower form:
      a00 a11 a22 a33 a44 a55
      a10 a21 a32 a43 a54 *
      a20 a31 a42 a53 *   *

  Cells marked with * are not used.

  Parameters:
    a_band(u+1, M) array_like

      The bands of the M by M matrix a.

    lower bool, optional

      Is the matrix in the lower form. (Default is upper form)

    eigvals_only bool, optional

      Compute only the eigenvalues and no eigenvectors. (Default: calculate also eigenvectors)

    overwrite_a_band bool, optional

      Discard data in a_band (may enhance performance)

    select{‘a’, ‘v’, ‘i’}, optional

      Which eigenvalues to calculate

      select | calculated
      --- | ---
      ‘a’ | All eigenvalues
      ‘v’ | Eigenvalues in the interval (min, max]
      ‘i’ | Eigenvalues with indices min <= i <= max

    select_range(min, max), optional

      Range of selected eigenvalues

    max_ev int, optional

      For select==’v’, maximum number of eigenvalues expected. For other values of select, has no meaning.

      In doubt, leave this parameter untouched.

    check_finite bool, optional

      Whether to check that the input matrix contains 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.

  Returns:
    w(M,) ndarray

      The eigenvalues, in ascending order, each repeated according to its multiplicity.

    v(M, M) float or complex ndarray

      The normalized eigenvector corresponding to the eigenvalue w[i] is the column v[:,i].

  Raises:
    LinAlgError

      If eigenvalue computation does not converge.

  See also

  `eigvals_banded`

    eigenvalues for symmetric/Hermitian band matrices

  `eig`

    eigenvalues and right eigenvectors of general arrays.

  `eigh`

    eigenvalues and right eigenvectors for symmetric/Hermitian arrays

  `eigh_tridiagonal`

    eigenvalues and right eigenvectors for symmetric/Hermitian tridiagonal matrices

  Examples

      >>> import numpy as np
      >>> from scipy.linalg import eig_banded
      >>> A = np.array([[1, 5, 2, 0], [5, 2, 5, 2], [2, 5, 3, 5], [0, 2, 5, 4]])
      >>> Ab = np.array([[1, 2, 3, 4], [5, 5, 5, 0], [2, 2, 0, 0]])
      >>> w, v = eig_banded(Ab, lower=True)
      >>> np.allclose(A @ v - v @ np.diag(w), np.zeros((4, 4)))
      True
      >>> w = eig_banded(Ab, lower=True, eigvals_only=True)
      >>> w
      array([-4.26200532, -2.22987175,  3.95222349, 12.53965359])

  Request only the eigenvalues between `[-3, 4]`

      >>> w, v = eig_banded(Ab, lower=True, select='v', select_range=[-3, 4])
      >>> w
      array([-2.22987175,  3.95222349])