msmtools.analysis.rdl_decomposition¶

msmtools.analysis.rdl_decomposition(T, k=None, norm='auto', ncv=None, reversible=False, mu=None)

Compute the decomposition into eigenvalues, left and right eigenvectors.

Parameters:
• T ((M, M) ndarray or sparse matrix) – Transition matrix
• k (int (optional)) – Number of eigenvector/eigenvalue pairs
• norm ({'standard', 'reversible', 'auto'}, optional) –

which normalization convention to use

norm
’standard’ LR = Id, is a probabilitydistribution, the stationary distributionof T. Right eigenvectors Rhave a 2-norm of 1
’reversible’ R and L are related via L[0, :]*R
’auto’ reversible if T is reversible, else standard.
• 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
Returns:

• R ((M, M) ndarray) – The normalized (“unit length”) right eigenvectors, such that the column R[:,i] is the right eigenvector corresponding to the eigenvalue w[i], dot(T,R[:,i])=w[i]*R[:,i]
• D ((M, M) ndarray) – A diagonal matrix containing the eigenvalues, each repeated according to its multiplicity
• L ((M, M) ndarray) – The normalized (with respect to R) left eigenvectors, such that the row L[i, :] is the left eigenvector corresponding to the eigenvalue w[i], dot(L[i, :], T)=w[i]*L[i, :]

Examples

>>> import numpy as np
>>> from msmtools.analysis import rdl_decomposition
>>> T = np.array([[0.9, 0.1, 0.0], [0.5, 0.0, 0.5], [0.0, 0.1, 0.9]])
>>> R, D, L = rdl_decomposition(T)

Matrix with right eigenvectors as columns

>>> R
array([[  1.00000000e+00,   1.04880885e+00,   3.16227766e-01], ...

Diagonal matrix with eigenvalues

>>> D
array([[ 1.0+0.j,  0.0+0.j,  0.0+0.j],
[ 0.0+0.j,  0.9+0.j,  0.0+0.j],
[ 0.0+0.j,  0.0+0.j, -0.1+0.j]])

Matrix with left eigenvectors as rows

>>> L # +doctest: +ELLIPSIS
array([[  4.54545455e-01,   9.09090909e-02,   4.54545455e-01], ...