Find the stationary distribution

[lwk99v]

hi here is the question and i don't know how to solve it.
a transition matrix P is called doubly stochastic if not only its rows sum up to one, but also its columns. In exact terms, P=(pij) which i,j is the elements of E is called doubly stochastic if
pij is greater or equal to 0 and the sum of pik=1 and the sum of pkj=1
for all i,j elements E.
and X=(Xn:n is elements of natural number) be an irreducible homogeneous Markov chain with a doubly stochastic transition matrix P. Assume that the state space E is finite. Determine the stationary distribution for X.

So how could i actually solve this question?please help and many thanks.

 

[Valkarie]

The stationary distribution for an irreducible homogeneous Markov chain with a doubly stochastic transition matrix P can be determined by solving the following system of linear equations:p11*pi1 + p12*pi2 + ... + p1n*pin = pi1 p21*pi1 + p22*pi2 + ... + p2n*pin = pi2 ...pn1*pi1 + pn2*pi2 + ... + pnn*pin = pinwhere pi1, pi2, ..., pin are the probabilities associated with each state. The solution to this system of equations will give the stationary distribution of the chain.