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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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Try the WP page on Perron-Frobenius theorem.