Skip to main content
`Ctrl`+`K`
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.

  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]])