Q:

Does a nonnegative real matrix always have a nonnegative eigenvector?

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?

A:

Try the WP page on Perron-Frobenius theorem.

