O Teorema de Perron-Frobenius

Abstract

This work aimed to study the spectrum of a certain class of matrices. More specifically, we present a detailed proof of the Perron-Frobenius theorem in the context of this class of matrices, following the method developed by Weilandt. This is a research that has as its methodology the bibliographic review. Perron’s theorem was stated and proved in 1907, which guarantees the existence of a maximal eigenvalue and a strictly positive associated eigenvector, for the class of square matrices with positive entries. In 1912, Frobenius extended this result to the class of non-negative and irreducible matrices. The classical form of the theorem is presented in three which state the existence of a maximal eigenvalue r, the existence of an eigenvector v with all positive entries associated with r and the influence of the variation of A on the variation of r. In addition to the developed theory presenting interesting results, this theorem is very important in several applications in areas such as economics, demography, physics, probability, among others.