Continuity of the Perron Root

TL;DR

This paper provides a simple, real-analysis-based proof that the Perron root of a nonnegative matrix varies continuously with the matrix entries, avoiding complex analysis.

Contribution

It offers a new, self-contained real analysis proof of the continuity of the Perron root, previously shown via complex eigenvalue theorems.

Findings

Perron root varies continuously with matrix entries

Proof relies solely on real analysis principles

Avoids complex number arguments

Abstract

That the Perron root of a square nonnegative matrix A varies continuously with the entries in A is a corollary of theorems regarding continuity of eigenvalues or roots of polynomial equations, the proofs of which necessarily involve complex numbers. But since continuity of the Perron root is a question that is entirely in the field of real numbers, it seems reasonable that there should exist a development involving only real analysis. This article presents a simple and completely self-contained development that depends only on real numbers and first principles.