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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.

  The documentation is written assuming array arguments are of specified “core” shapes. However, array argument(s) of this function may have additional “batch” dimensions prepended to the core shape. In this case, the array is treated as a batch of lower-dimensional slices; see Batched Linear Operations for details. Note that calls with zero-size batches are unsupported and will raise a `ValueError`.

  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