What exactly is the Perron–Frobenius theorem? In different books and papers I read different statements, and I don't know what the truth is. In Wikipedia there are also a lot of statements under this label.
And if somebody can characterize the statement for me, then I need also a proof. I read that the proof needs Brouwer's fixed point theorem, but is there a quite easy way to prove it?
If someone refers to the "Perron-Frobenius theorem" then you cannot be sure exactly what they mean. In practice the term indicates some subset of the union of 1.2.2 and 1.3 on the wikipedia page. If you want to refer to it, the safest thing is to cite it as stated in some reasonable text on linear algebra (Horn and Johnson, to give just one example).
The hardest part of the proof is showing that a positive matrix has a positive eigenvector, and this is where one might use Brouwer's fixed point theorem. I would say that there is not a "quite easy way" to prove this, but certainly there are proofs that do not use Brouwer.
