Two-Dimensional Projective Collapse and Sharp Distortion Bounds for Products of Positive Matrices

Eugene Kritchevski

Paper

Two-Dimensional Projective Collapse and Sharp Distortion Bounds for Products of Positive Matrices

Abstract

We introduce an elementary framework that captures the mechanism driving the alignment of rows and columns in products of positive matrices. All worst-case misalignment occurs already in dimension two, leading to an explicit collapse principle and a sharp nonlinear bound for finite products. The proof avoids Hilbert-metric and cone-theoretic techniques, relying instead on basic calculus. In the Hilbert metric, the classical Birkhoff-Bushell contraction captures only the linearized asymptotic regime, whereas our nonlinear envelope function gives the exact worst-case behavior for finite products.