# Primitive matrices has positive powers.

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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That depends on your definition. Usually this is the definition of a primitive matrix. What is your definition? –  Dennis Gulko Jan 24 '12 at 14:46
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. –  Zizo Jan 24 '12 at 14:59