When can a matrix with negative entries have a completely non-negative dominant eigenvector? Perron-Forbenius obviously answers this question for positive and for certain non-negative matrices. I want to know whether these conditions can be weakened at all. In other words, what, if anything, can be said about the class of matrices which have some negative entries but still have non-negative dominant eigenvectors. Is this even possible? I haven't been able to construct an example although I have not been trying that for long.
 A: Let $J$ be the all-ones matrix, order $n\times n$ and set $M=J-2I$.
The largest eigenvalue is $n-2$ with the all-ones vector as the eigenvector, all other eigenvalues are $-2$. So assume $n\ge5$.
For an alternative, take your favourite non-negative irreducible matrix $M$ and let $z$ be the eigenvector for the dominant eigenvalue with norm 1. Let $L$ be an orthogonal matrix whose columns are an orthonormal basis that contains $z$. Then $L^TML$ has the same eigenvalues as $M$ and the eigenvector belonging to the dominant eigenvalue is $z$. But in general $L^TML$
will not be non-negative.
So it's hard to see what a useful answer to your question would look like.
A: Basically, you are looking for matrices with the so-called "generalized Perron-Frobenius property" (there is a weak and strong version, of course, and the nuances of terminology somewhat differ in various authors). In recent years they have been studied quite a lot and - very roughly - this is equivalent to being eventually nonnegative (nonegative for some integer power).
See, for example these papers for more information, careful statements of results and examples galore:
http://www.math.uoi.gr/~dnoutsos/Noutsos_LAA_2005.pdf
http://ftp.gwdg.de/pub/misc2/EMIS/journals/ELA/ela-articles/articles/vol17_pp389-413.pdf
http://www.sciencedirect.com/science/article/pii/S002437950200366X
