How do I find out eigenvectors corresponding to a particular eigenvalue?
I have a stochastic matrix(P), one of the eigenvalues of which is 1. I need to find the eigenvector corresponding to the eigenvalue 1.
The scipy function scipy.linalg.eig returns the array of eigenvalues and eigenvectors.
D, V = scipy.linalg.eig(P) Here D(array of values) and V(array of vectors) are both vectors.
One way is to do a search in D and extract the corresponding eigenvector in V. Is there an easier way?
If you are looking for one eigenvector corresponding to one eigenvalue, it could be much more efficient to use the scipy.sparse.linalg implementation of the eig function. It allows to look for a fixed number of eigenvectors and to shift the search around a specific value. You could do for instance :
values, vectors = scipy.sparse.linalg.eigs(P, k=1, sigma=1) import numpy as np import numpy.linalg as linalg P = np.array([[2, 0, 0], [0, 1, 0], [0, 0, 3]]) D, V = linalg.eig(P) print(D) # [ 2. 1. 3.] The eigenvectors are columns of V:
V = V.T for val, vec in zip(D, V): assert np.allclose(np.dot(P, vec), val*vec) So the eigenvector corresponding to eigenvalue 1.0 is
def near(a, b, rtol = 1e-5, atol = 1e-8): return np.abs(a-b)<(atol+rtol*np.abs(b)) print(V[near(D, 1.0)]) # [[ 0. 1. 0.]] Since there can be more than one eigenvector with the same eigenvalue, V[near(D, 1.0)] returns a 2-dimensional array -- each row of the array is an eigenvector with an eigenvalue of 1.0.