scipy.linalg.cdf2rdf

scipy.linalg.cdf2rdf(w, v)

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.

New 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.

New in version 1.1.0.

Examples

>>> 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]])