The Perron-Frobenius theorem is a powerful and useful result about the eigenvalues and eigenvectors of a non-negative matrix. I will not be proving the theorem but will instead focus on its applications. In particular, I will discuss how it can be used to understand the behaviour of a certain class of dynamical systems, namely certain systems of first-order ordinary differential equations where the couplings between variables are specified by a graph. Some familiarity with graph theory will be useful, but I will try to recapitulate all the basic concepts needed for this talk.

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Last updated: 14 Feb 2019