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

Returns:

is_reversible – True, if T is reversible, False otherwise

Return type:

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