Strongly connected primitive matrix

[Doom of Doom]

If H is a nxn primitive, irreducible matrix, is it always true that H^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?

 

[Doom of Doom]

Ok, I thought about it a little more. H^n-1 might not necessarily be positive, since it might not be possible to go from i to j in exactly n-1 steps, but it would be possible for some k, 1<k<n-1. Also, the diagonal entries of H^n-1 would not necessarily be positive (if for some node you have to hit every other node before coming back to the original node

Thus, I + H + H² +... + H^n-1 should have all positive entries.

i.e., if the digraph is strongly positive, it should be possible to get from every node to every other node in at most n-1 steps. Thus, for all i,j, with i$$\neq$$j, (H^k)_ij must be greater than zero for some k, 1<k<n-1.

How can I prove THIS?

thanks