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

>>> 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