scipy.linalg.

cdf2rdf#

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

Complex diagonal form to real diagonal block form.

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.

Array argument(s) of this function may have additional “batch” dimensions prepended to the core shape. In this case, the array is treated as a batch of lower-dimensional slices; see Batched Linear Operations for details.

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]])