Sharp Finite-Time Distortion Bounds for Products of Positive Matrices

TL;DR

This paper establishes precise, dimension-independent bounds on how much the rows and columns of products of positive matrices can deviate from proportionality, revealing that the worst-case misalignment is already evident in two-dimensional cases and follows a specific Mobius law.

Contribution

The paper introduces a sharp, dimension-free bound on finite-time distortion in products of positive matrices, refining classical contraction theory with an explicit Mobius law.

Findings

Worst-case misalignment is captured in dimension two.

The bounds follow an explicit Mobius law.

Refines classical Birkhoff-Bushell contraction theory.

Abstract

We study the deviation from proportionality of rows and columns in products of positive matrices. We prove a sharp, dimension-free bound showing that worst-case misalignment is already captured in dimension two and follows an explicit Mobius law, refining the classical Birkhoff-Bushell contraction theory.