What are some cool implications of the Perron-Fronenius Theorem? I’m doing a 15 minute presentation on The Perron-Frobenius Theorem. 
Does anyone know any nice results or examples I can include in it?
 A: This is cliché, but Perron-Frobenius theorem is what guarantees that Google's earliest version of PageRank algorithm can produce a unique page ranking.
Roughly speaking, in the earliest version of PageRank algorithm, the page ranks of the webpages that have mentioned a search term are obtained from some specific eigenvector of the stochastic matrix $P$ containing the conditional probabilities $p_{ij}$ that a reader of page $j$ will visit page $i$ provided that he/she will do so by clicking a link to page $i$ on page $j$. By Perron-Frobenius theorem for nonnegative matrices, the spectral radius $\rho(P)$ is always an eigenvalue of $P$. Now the idea is to define the page rank of webpage $i$ as the $i$-th element in the eigenvector for $\rho(P)$. This is called "eigen ranking" in some literature.
Unfortunately, this doesn't work, because the eigenspace for $\rho(P)$ may be two or higher dimensional, so that the eigenvectors are not unique up to scaling.
However, by modifying $P$ to the convex combination $(1-t)P+t\frac{E}{n}$ (where $0<t<1$ is called a "damping factor", $E$ is the $n\times n$ all-one matrix and $n$ is the number of webpages that mention the relevant search term and that the web crawler can reach), the new $P$ becomes positive. (Intuitively, the introduction of $t\frac{E}{n}$ means that there is a probability $t$ that a user may switch from one webpage to another at random.) Hence, by Perron-Frobenius theorem for positive matrices, $\rho(P)$ becomes a simple eigenvalue and it has a unique eigenvector (up to scaling). This allows one to rank the webpages by the values of their corresponding entries in this eigenvector. Although the page rank of a webpage may become different if one chooses a different value of the parameter $t$, if one keeps $t$ fixed, there is at least a consistent ordering of the page ranks.
A: In physics an application of Perron-Frobenius (PF) theorem due to Mattis allows to show that, at finite size and for bipartite lattices, the ground state of the antiferromagnetic Heisenberg model in D dimension is unique. 
Matrices satisfying PF are also important in the field of quantum computing and they are termed stoquastic. 
