hessenberg — SciPy v1.15.2 Manual (original) (raw)

scipy.linalg.

scipy.linalg.hessenberg(a, calc_q=False, overwrite_a=False, check_finite=True)[source]#

Compute Hessenberg form of a matrix.

The Hessenberg decomposition is:

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_qbool, optional

Whether to compute the transformation matrix. Default is False.

overwrite_abool, optional

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

check_finitebool, 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