Growth in products of matrices: fastest, average, and generic

TL;DR

This paper investigates the growth rates of products of 2x2 matrices, providing partial and complete answers to maximum entry growth, expected maximum entry in random products, and an upper bound for the Lyapunov exponent in nonnegative cases.

Contribution

It offers new bounds and methods for analyzing the growth of matrix products, especially in the context of random matrix sequences.

Findings

Partial answer to maximum entry growth problem.

Complete characterization of expected maximum entry in random products.

Simple method for upper bounding Lyapunov exponent with nonnegative matrices.

Abstract

The problems that we consider in this paper are as follows. Let A and B be 2x2 matrices (over reals). Let w(A, B) be a word of length n. After evaluating w(A, B) as a product of matrices, we get a 2x2 matrix, call it W. What is the largest (by the absolute value) possible entry of W, over all w(A, B) of length n, as a function of n? What is the expected absolute value of the largest (by the absolute value) entry in a random product of n matrices, where each matrix is A or B with probability 0.5? What is the Lyapunov exponent for a random matrix product like that? We give partial answer to the first of these questions and an essentially complete answer to the second question. For the third question (the most difficult of the three), we offer a very simple method to produce an upper bound on the Lyapunov exponent in the case where all entries of the matrices A and B are nonnegative.