eigenmarkov
eigenmarkov — normalized left and right Markov eigenvectors
Calling sequence
[M,Q]=eigenmarkov(P)
Parameters
| P | : real N x N Markov matrix. Sum of entries in each row should add to one. |
| M | : real matrix with N columns. |
| Q | : real matrix with N rows. |
Description
Returns normalized left and right eigenvectors
associated with the eigenvalue 1 of the Markov transition matrix P.
If the multiplicity of this eigenvalue is m and P
is N x N, M is a m x N matrix and Q a N x m matrix.
M(k,:) is the probability distribution vector associated with the kth
ergodic set (recurrent class). M(k,x) is zero if x is not in the
k-th recurrent class.
Q(x,k) is the probability to end in the k-th recurrent class starting
from x. If P^k converges for large k (no eigenvalues on the
unit circle except 1), then the limit is Q*M (eigenprojection).
Examples
//P has two recurrent classes (with 2 and 1 states) 2 transient states
P=genmarkov([2,1],2)
[M,Q]=eigenmarkov(P);
P*Q-Q
Q*M-P^20
See also
genmarkov, classmarkov
