function eigenvectors¶
- deeptime.markov.tools.analysis.eigenvectors(T, k=None, right=True, ncv=None, reversible=False, mu=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
right (bool, optional) – If right=True compute right eigenvectors, left eigenvectors otherwise
reversible (bool, optional) – Indicate that transition matrix is reversible
mu ((M,) ndarray, optional) – Stationary distribution of T
- 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.
- Return type:
numpy.ndarray, shape=(d, n)
See also
Notes
Eigenvectors are computed using the scipy interface to the corresponding LAPACK/ARPACK routines.
If reversible=False, 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.
If reversible=True the the eigenvectors of the similar symmetric matrix sqrt(mu_i / mu_j) p_{ij} will be used to compute the eigenvectors of T.
The precomputed stationary distribution will only be used if reversible=True.
Examples
>>> import numpy as np >>> from deeptime.markov.tools.analysis import eigenvectors
>>> T = np.array([[0.9, 0.1, 0.0], [0.5, 0.0, 0.5], [0.0, 0.1, 0.9]]) >>> R = eigenvectors(T)
Matrix with right eigenvectors as columns
>>> R array([[ 5.77350269e-01, 7.07106781e-01, 9.90147543e-02]...