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How to prove this statement:

If $P = [p_{ij}]_{ 1 \leqslant i,j \leqslant m} \geqslant 0$ is a primitive matrix, then there exists a $k \in \mathbb{N}$ such that $ P^{k} > 0.$ Moreover $ P^{k+i}>0$ for all $ i = 1,2, \dots $

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    I use the definition appeared in (Roger A. Horn and Charles A. Johnson. Matrix analysis. Cambridge University Press, 1985" page:516) which is a matrix $ P = [p_{ij}]_{1 \leqslant i,j \leqslant m} \geqslant 0$ is called primitive if it is irreducible matrix and it has a strictly dominant eigenvalue.2012-01-24

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