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.

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`

`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.

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]])
```