Perron-Frobenius' Theory and Applications

Abstract: This is a literature study, in linear algebra, about positive and nonnegative matrices and their special properties. We say that a matrix or a vector is positive/nonnegative if all of its entries are positive/nonnegative. First, we study some generalities and become acquainted with two types of nonnegative matrices; irreducible and reducible. After exploring their characteristics we investigate and prove the two main theorems of this subject, namely Perron's and Perron-Frobenius' theorem. In short Perron's theorem from 1907 tells us that the spectral radius of a positive matrix is a simple eigenvalue of the matrix and that its eigenvector can be taken to be positive. In 1912, Georg Frobenius generalized Perron's results also to irreducible nonnegative matrices. The two theorems have a wide range of applications in both pure mathematics and practical matters. In real world scenarios, many measurements are nonnegative (length, time, amount, etc.) and so their mathematical formulations often relate to Perron-Frobenius theory. The theory's importance to linear dynamical systems, such as Markov chains, cannot be overstated; it determines when, and to what, an iterative process will converge. This result is in turn the underlying theory for the page-ranking algorithm developed by Google in 1998. We will see examples of all these applications in chapters four and five where we will be particularly interested in different types of Markov chains.  The theory in this thesis can be found in many books. Here, most of the material is gathered from Horn-Johnson [5], Meyer [9] and Shapiro [10]. However, all of the theorems and proofs are formulated in my own way and the examples and illustrations are concocted by myself, unless otherwise noted.