eigenmarkov - normalized left and right Markov eigenvectors
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).
//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
genmarkov , classmarkov ,