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:

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 (xx.conjugate() <= 1.0) 'ouc' Outside the unit circle (xx.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: The algorithm failed due to a failure of the QR algorithm to compute all eigenvalues If eigenvalue sorting was requested, the eigenvalues could not be reordered due to a failure to separate eigenvalues, usually because of poor conditioning If eigenvalue sorting was requested, roundoff errors caused the leading eigenvalues to no longer satisfy the sorting condition

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:

The algorithm failed due to a failure of the QR algorithm to compute all eigenvalues

If eigenvalue sorting was requested, the eigenvalues could not be reordered due to a failure to separate eigenvalues, usually because of poor conditioning

If eigenvalue sorting was requested, roundoff errors caused the leading eigenvalues to no longer satisfy the sorting condition