cdf2rdf — SciPy v1.15.2 Manual (original) (raw)

scipy.linalg.

scipy.linalg.cdf2rdf(w, v)[source]#

Converts complex eigenvalues w and eigenvectors v to real eigenvalues in a block diagonal form wr and the associated real eigenvectors vr, such that:

continues to hold, where X is the original array for which w andv are the eigenvalues and eigenvectors.

Added in version 1.1.0.

Parameters:

w(…, M) array_like

Complex or real eigenvalues, an array or stack of arrays

Conjugate pairs must not be interleaved, else the wrong result will be produced. So [1+1j, 1, 1-1j] will give a correct result, but [1+1j, 2+1j, 1-1j, 2-1j] will not.

v(…, M, M) array_like

Complex or real eigenvectors, a square array or stack of square arrays.

Returns:

wr(…, M, M) ndarray

Real diagonal block form of eigenvalues

vr(…, M, M) ndarray

Real eigenvectors associated with wr

See also

eig

Eigenvalues and right eigenvectors for non-symmetric arrays

rsf2csf

Convert real Schur form to complex Schur form

Notes

w, v must be the eigenstructure for some real matrix X. For example, obtained by w, v = scipy.linalg.eig(X) orw, v = numpy.linalg.eig(X) in which case X can also represent stacked arrays.

Added in version 1.1.0.

Examples

import numpy as np X = np.array([[1, 2, 3], [0, 4, 5], [0, -5, 4]]) X array([[ 1, 2, 3], [ 0, 4, 5], [ 0, -5, 4]])

from scipy import linalg w, v = linalg.eig(X) w array([ 1.+0.j, 4.+5.j, 4.-5.j]) v array([[ 1.00000+0.j , -0.01906-0.40016j, -0.01906+0.40016j], [ 0.00000+0.j , 0.00000-0.64788j, 0.00000+0.64788j], [ 0.00000+0.j , 0.64788+0.j , 0.64788-0.j ]])

wr, vr = linalg.cdf2rdf(w, v) wr array([[ 1., 0., 0.], [ 0., 4., 5.], [ 0., -5., 4.]]) vr array([[ 1. , 0.40016, -0.01906], [ 0. , 0.64788, 0. ], [ 0. , 0. , 0.64788]])

vr @ wr array([[ 1. , 1.69593, 1.9246 ], [ 0. , 2.59153, 3.23942], [ 0. , -3.23942, 2.59153]]) X @ vr array([[ 1. , 1.69593, 1.9246 ], [ 0. , 2.59153, 3.23942], [ 0. , -3.23942, 2.59153]])