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**akbar** let $\displaystyle P$ be a stochastic matrix, $\displaystyle \pi$ a non-zero line vector solution of the equation $\displaystyle \pi P = \pi$ (the solution necessarily exists, since $\displaystyle \det(P-I)=\det(P^T-I)=0$).

If $\displaystyle \pi = (\pi_1,...,\pi_n)$, how do you prove that $\displaystyle (|\pi_1|,...,|\pi_n|)$ is also solution?

I was wondering if it is possible to obtain the result without too elaborate references to ergodicity results and state subsets decomposition.

Thanks for your help.