msmtools.analysis.eigenvalues(T, k=None, ncv=None, reversible=False, mu=None)

Find eigenvalues of the transition matrix.

  • T ((M, M) ndarray or sparse matrix) – Transition matrix
  • k (int (optional)) – Compute the first k eigenvalues of T
  • ncv (int (optional)) – The number of Lanczos vectors generated, ncv must be greater than k; it is recommended that ncv > 2*k
  • reversible (bool, optional) – Indicate that transition matrix is reversible
  • mu ((M,) ndarray, optional) – Stationary distribution of T

w – Eigenvalues of T. If k is specified, w has shape (k,)

Return type:

(M,) ndarray


Eigenvalues are returned in order of decreasing magnitude.

If reversible=True the the eigenvalues of the similar symmetric matrix sqrt(mu_i / mu_j) p_{ij} will be computed.

The precomputed stationary distribution will only be used if reversible=True.


>>> import numpy as np
>>> from msmtools.analysis import eigenvalues
>>> T = np.array([[0.9, 0.1, 0.0], [0.5, 0.0, 0.5], [0.0, 0.1, 0.9]])
>>> w = eigenvalues(T)
>>> w
array([ 1.0+0.j, 0.9+0.j, -0.1+0.j])