pyemma.msm.analysis.eigenvectors

pyemma.msm.analysis.eigenvectors(T, k=None, right=True, ncv=None)

Compute eigenvectors of given transition matrix.

Parameters:
  • T (numpy.ndarray, shape(d,d) or scipy.sparse matrix) – Transition matrix (stochastic matrix)
  • k (int (optional)) – Compute the first k eigenvectors
  • ncv (int (optional)) – The number of Lanczos vectors generated, ncv must be greater than k; it is recommended that ncv > 2*k
Returns:

eigvec – The eigenvectors of T ordered with decreasing absolute value of the corresponding eigenvalue. If k is None then n=d, if k is int then n=k.

If right = True, the right eigenvectors are returned as column vectors. If right = False, the left eigenvectors are returned as row vectors
Return type:

numpy.ndarray, shape=(d, n)

See also

rdl_decomposition()

Notes

Eigenvectors are computed using the scipy interface to the corresponding LAPACK/ARPACK routines.

The returned eigenvectors \(v_i\) are normalized such that

\[\langle v_i, v_i \rangle = 1\]

This is the case for right eigenvectors \(r_i\) as well as for left eigenvectors \(l_i\).

If you desire orthonormal left and right eigenvectors please use the rdl_decomposition method.

Examples

>>> from pyemma.msm.analysis import eigenvalues

>>> T = np.array([[0.9, 0.1, 0.0], [0.5, 0.0, 0.5], [0.0, 0.1, 0.9]])
>>> R = eigenvalues(T)

Matrix with right eigenvectors as columns

>>> R
array([[  5.77350269e-01,   7.07106781e-01,   9.90147543e-02],
       [  5.77350269e-01,  -5.50368425e-16,  -9.90147543e-01],
       [  5.77350269e-01,  -7.07106781e-01,   9.90147543e-02]])