# Perron-Frobenius Theorem

## Table of Contents

## 1 Definition

An \( n \times n \) matrix \( A \) is said to be **irreducible** if for all pairs \( (i, j) \) there exists some \( k \in \mathbb{Z} \) such that \( M_{i, j}^k > 0 \).

We say an \( n \times n \) matrix \( A \) is **primitive** if there exists some \( k \) for which all entries of \( A^k \) are positive. We can see that primitivity is a similar yet stronger condition than irreducibility.

## 2 Perron-Frobenius Theorem

If \( A \) is an irreducible matrix then \( A \) has an eigenvalue \( \lambda_0 \) satisfying:

- All other eigenvalues of \( A \) satisfy \( |\lambda| < \lambda_0 \)
- \( \lambda_0 \) has algebraic and geometric multiplicity equal to one
- \( \lambda_0 \) has an eigenvector with all entries greater than zero
- Further, any non-negative eigenvector of \( A \) is a multiple of this eigenvector