The Perron–Frobenius Theorem

By James R. Kirkwood, B. Kirkwood. Linear Algebra, 2020

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The theorem we will discuss in this chapter (to be stated below) about matrices with non-negative entries, was proved, for matrices with strictly positive entries, by Oskar Perron (1880-1975) in 1907 and extended by Ferdinand Georg Frobe-nius (1849-1917) to matrices which have non-negative entries and are irreducible (definition below) in 1912. This theorem has miriads of applications, several of which we will study in this book. 9.1 Non-negative and positive matrices. We begin with some definitions. We say that a real matrix T is non-negative (or positive) if all the entries of T are non-negative (or positive). We write T ≥ 0 or T > 0. We will use these definitions primarily for square (n × n) matrices and for column vectors = (n × 1) matrices, although rectangular matrices will come into the picture at one point.

Subfields: Algebra and Number Theory

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