scipy.linalg.

schur#

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

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

When the dtype of a is real, this specifies whether to compute the real or complex Schur decomposition. When the dtype of a is complex, this argument is ignored, and the complex Schur decomposition is computed.

lworkint, optional

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

overwrite_abool, 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 an eigenvalue, returns a boolean denoting whether the eigenvalue should be sorted to the top-left (True).

  • If output='complex' OR the dtype of a is complex, the callable should have one argument: the eigenvalue expressed as a complex number.

  • If output='real' AND the dtype of a is real, the callable should have two arguments: the real and imaginary parts of the eigenvalue, respectively.

Alternatively, string parameters may be used:

'lhp'   Left-hand plane (real(eigenvalue) < 0.0)
'rhp'   Right-hand plane (real(eigenvalue) >= 0.0)
'iuc'   Inside the unit circle (abs(eigenvalue) <= 1.0)
'ouc'   Outside the unit circle (abs(eigenvalue) > 1.0)

Defaults to None (no sorting).

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:
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.

sdimint

If and only if sorting was requested, a third return value will contain the number of eigenvalues satisfying the sort condition. Note that complex conjugate pairs for which the condition is true for either eigenvalue count as 2.

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

>>> import numpy as np
>>> 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], # may vary
       [ 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])   # may vary

A custom eigenvalue-sorting condition that sorts by positive imaginary part is satisfied by only one eigenvalue.

>>> _, _, sdim = schur(A, output='complex', sort=lambda x: x.imag > 1e-15)
>>> sdim
1

When output='real' and the array a is real, the sort callable must accept the real and imaginary parts as separate arguments. Note that now the complex eigenvalues -0.32948354+0.80225456j and -0.32948354-0.80225456j will be treated as a complex conjugate pair, and according to the sdim documentation, complex conjugate pairs for which the condition is True for either eigenvalue increase sdim by two.

>>> _, _, sdim = schur(A, output='real', sort=lambda x, y: y > 1e-15)
>>> sdim
2
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