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Of course the term "non-negative" is entry-wise.

I know the fact that if this matrix is a stochastic matrix, that is, the sum of each of its rows is 1, then it has a stationary probability vector $\pi$, which is of course non-negative.

But what if it is just a general non-negative matrix? Does it again always have a non-negative eigenvector?

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I see. You deleted your previous question on this. The answer is yes. – Will Jagy Sep 19 '12 at 20:07
@WillJagy: Yeah. Sorry about that. I tried really hard to analyze it, and did some experiments, but just can't see why it is true. – Voldemort Sep 19 '12 at 20:09
Try this. – Did Sep 19 '12 at 20:15
@did: Oh! Thanks. How silly I was. That's just the theorem that proved the existence of stationary probability vector. – Voldemort Sep 19 '12 at 20:18
This is knpwn as the Perron_frobenius theorem, there is a WP page: – kjetil b halvorsen Sep 19 '12 at 22:42
up vote 3 down vote accepted

Try the WP page on Perron-Frobenius theorem.

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