If H is a nxn primitive, irreducible matrix, is it always true that H(adsbygoogle = window.adsbygoogle || []).push({}); ^{n-1}> 0? That is, every entry in H^{n-1}is positive.

From my class notes, the definition of H primitive is that there exists some k>0 such that H^{k}> 0. And a matrix is irreducible if its digraph is strongly connected (that is, it is possible to reach every node from every other node in a finite number of steps).

I am interpreting H as a matrix of a strongly connected digraph. Since it is strongly connected, it should be possible to reach any other node after n-1 steps, and thus (H^{n -1})_{ij}should be positive since there is a positive probability that it will get to node j from node i in n-1 steps.

Is there a good way to prove that every element of H^{n-1}must be positive?

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# Strongly connected primitive matrix

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