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# scipy.linalg.hessenberg

scipy.linalg.hessenberg(a, calc_q=False, overwrite_a=False, check_finite=True)

  Compute Hessenberg form of a matrix.

  The Hessenberg decomposition is:

      A = Q H Q^H

  where Q is unitary/orthogonal and H has only zero elements below the first sub-diagonal.

  Parameters:
    a(M, M) array_like

      Matrix to bring into Hessenberg form.

    calc_q bool, optional

      Whether to compute the transformation matrix. Default is False.

    overwrite_a bool, optional

      Whether to overwrite a; may improve performance. Default is False.

    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:
    H(M, M) ndarray

      Hessenberg form of a.

    Q(M, M) ndarray

      Unitary/orthogonal similarity transformation matrix `A = Q H Q^H`. Only returned if `calc_q=True`.

  Examples

      >>> import numpy as np
      >>> from scipy.linalg import hessenberg
      >>> A = np.array([[2, 5, 8, 7], [5, 2, 2, 8], [7, 5, 6, 6], [5, 4, 4, 8]])
      >>> H, Q = hessenberg(A, calc_q=True)
      >>> H
      array([[  2.        , -11.65843866,   1.42005301,   0.25349066],
             [ -9.94987437,  14.53535354,  -5.31022304,   2.43081618],
             [  0.        ,  -1.83299243,   0.38969961,  -0.51527034],
             [  0.        ,   0.        ,  -3.83189513,   1.07494686]])
      >>> np.allclose(Q @ H @ Q.conj().T - A, np.zeros((4, 4)))
      True