Infinite products of nonnegative $2\times2$ matrices by nonnegative vectors

TL;DR

This paper characterizes the conditions under which infinite products of nonnegative 2x2 matrices, applied to a nonnegative vector, converge to a limit for all sequences avoiding the zero vector.

Contribution

It provides a complete necessary and sufficient criterion for the convergence of normalized matrix product sequences in the 2x2 nonnegative case.

Findings

Derived explicit convergence conditions for all matrix sequences avoiding zero vectors.

Established a characterization applicable to all nonnegative 2x2 matrix sets and initial vectors.

Enhanced understanding of the asymptotic behavior of matrix products in low dimensions.

Abstract

Given a finite set $\{M_0,\dots,M_{d-1}\}$ of nonnegative $2\times 2$ matrices and a nonnegative column-vector $V$, we associate to each $(\omega_n)\in\{0,\dots,d-1\}^\mathbb N$ the sequence of the column-vectors $\displaystyle{M_{\omega_1}\dots M_{\omega_n}V\over\Vert M_{\omega_1}\dots M_{\omega_n}V\Vert}$. We give the necessary and sufficient condition on the matrices $M_k$ and the vector $V$ for this sequence to converge for all \hbox{$(\omega_n)\in\{0,\dots,d-1\}^\mathbb N$} such that $\forall n,\ M_{\omega_1}\dots M_{\omega_n}V\ne\begin{pmatrix}0\\0\end{pmatrix}$.