On Norms of Iterations of {0,1}-Matrices

TL;DR

This paper characterizes the detailed behavior of the norms of iterated {0,1}-matrices, especially when their spectral radius equals one, providing a complete understanding of their growth patterns.

Contribution

It offers a comprehensive analysis of the asymptotic behavior of matrix norms for {0,1}-matrices based on their spectral radius, including the case when it equals one.

Findings

For ho(M)>1, |M^n| grows exponentially.

When ho(M)=1, |M^n| can be bounded or unbounded.

The paper fully characterizes the sequence behavior in all cases.

Abstract

Let M be a b*b nonzero {0,1}-matrix. Let \rho(M) be its spectral radius and let |M^n| be the norm of its n-th iteration. In the case \rho(M)>1, we see from the spectral radius formula that {|M^n|}_{n=1}^\infty tends to \infty exponentially as n to \infty. In the case \rho(M)=1, {|M^n|}_{n=1}^\infty can be bounded or tend to \infty depending on M. The fine behavior of this sequence is completely characterized in the present paper.