scipy.linalg.cdf2rdf#

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:

vr @ wr = X @ vr

continues to hold, where X is the original array for which w and v 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) or w, 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]])