# msmtools.analysis.is_reversible¶

msmtools.analysis.is_reversible(T, mu=None, tol=1e-12)

Check reversibility of the given transition matrix.

Parameters: T ((M, M) ndarray or scipy.sparse matrix) – Transition matrix mu ((M,) ndarray (optional)) – Test reversibility with respect to this vector tol (float (optional)) – Floating point tolerance to check with is_reversible – True, if T is reversible, False otherwise bool

Notes

A transition matrix $$T=(t_{ij})$$ is reversible with respect to a probability vector $$\mu=(\mu_i)$$ if the follwing holds,

$\mu_i \, t_{ij}= \mu_j \, t_{ji}.$

In this case $$\mu$$ is the stationary vector for $$T$$, so that $$\mu^T T = \mu^T$$.

If the stationary vector is unknown it is computed from $$T$$ before reversibility is checked.

A reversible transition matrix has purely real eigenvalues. The left eigenvectors $$(l_i)$$ can be computed from right eigenvectors $$(r_i)$$ via $$l_i=\mu_i r_i$$.

Examples

>>> import numpy as np
>>> from msmtools.analysis import is_reversible

>>> P = np.array([[0.8, 0.1, 0.1], [0.5, 0.0, 0.5], [0.0, 0.1, 0.9]])
>>> is_reversible(P)
False

>>> T = np.array([[0.9, 0.1, 0.0], [0.5, 0.0, 0.5], [0.0, 0.1, 0.9]])
>>> is_reversible(T)
True