scipy.linalg.schur

scipy.linalg.schur(a, output='real', lwork=None, overwrite_a=False, sort=None, check_finite=True)

Compute Schur decomposition of a matrix.

The Schur decomposition is:

A = Z T Z^H

where Z is unitary and T is either upper-triangular, or for real Schur decomposition (output=’real’), quasi-upper triangular. In the quasi-triangular form, 2x2 blocks describing complex-valued eigenvalue pairs may extrude from the diagonal.

Parameters:

a : (M, M) array_like

Matrix to decompose

output : {‘real’, ‘complex’}, optional

Construct the real or complex Schur decomposition (for real matrices).

lwork : int, optional

Work array size. If None or -1, it is automatically computed.

overwrite_a : bool, optional

Whether to overwrite data in a (may improve performance).

sort : {None, callable, ‘lhp’, ‘rhp’, ‘iuc’, ‘ouc’}, optional

Specifies whether the upper eigenvalues should be sorted. A callable may be passed that, given a eigenvalue, returns a boolean denoting whether the eigenvalue should be sorted to the top-left (True). Alternatively, string parameters may be used:

'lhp'   Left-hand plane (x.real < 0.0)
'rhp'   Right-hand plane (x.real > 0.0)
'iuc'   Inside the unit circle (x*x.conjugate() <= 1.0)
'ouc'   Outside the unit circle (x*x.conjugate() > 1.0)

Defaults to None (no sorting).

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:

T : (M, M) ndarray

Schur form of A. It is real-valued for the real Schur decomposition.

Z : (M, M) ndarray

An unitary Schur transformation matrix for A. It is real-valued for the real Schur decomposition.

sdim : int

If and only if sorting was requested, a third return value will contain the number of eigenvalues satisfying the sort condition.

Raises:

LinAlgError

Error raised under three conditions:

1. The algorithm failed due to a failure of the QR algorithm to compute all eigenvalues
2. If eigenvalue sorting was requested, the eigenvalues could not be reordered due to a failure to separate eigenvalues, usually because of poor conditioning
3. If eigenvalue sorting was requested, roundoff errors caused the leading eigenvalues to no longer satisfy the sorting condition

See also

rsf2csf

Convert real Schur form to complex Schur form

Examples

>>> from scipy.linalg import schur, eigvals
>>> A = np.array([[0, 2, 2], [0, 1, 2], [1, 0, 1]])
>>> T, Z = schur(A)
>>> T
array([[ 2.65896708,  1.42440458, -1.92933439],
       [ 0.        , -0.32948354, -0.49063704],
       [ 0.        ,  1.31178921, -0.32948354]])
>>> Z
array([[0.72711591, -0.60156188, 0.33079564],
       [0.52839428, 0.79801892, 0.28976765],
       [0.43829436, 0.03590414, -0.89811411]])

>>> T2, Z2 = schur(A, output='complex')
>>> T2
array([[ 2.65896708, -1.22839825+1.32378589j,  0.42590089+1.51937378j],
       [ 0.        , -0.32948354+0.80225456j, -0.59877807+0.56192146j],
       [ 0.        ,  0.                    , -0.32948354-0.80225456j]])
>>> eigvals(T2)
array([2.65896708, -0.32948354+0.80225456j, -0.32948354-0.80225456j])

An arbitrary custom eig-sorting condition, having positive imaginary part, which is satisfied by only one eigenvalue

>>> T3, Z3, sdim = schur(A, output='complex', sort=lambda x: x.imag > 0)
>>> sdim
1