eigenmarkov normalized left and right Markov eigenvectors

## Syntax

    [M,Q]=eigenmarkov(P)

## Arguments

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