Minimax theorem for the spectral radius of the product of non-negative matrices

TL;DR

This paper establishes a minimax equality for the spectral radius of the product of non-negative matrices, under specific compactness and hourglass conditions, advancing theoretical understanding in matrix analysis.

Contribution

It proves a new minimax theorem for the spectral radius of matrix products with non-negative matrices satisfying the hourglass alternative, extending existing spectral theory.

Findings

Proves minimax equality for spectral radius of matrix products

Identifies conditions under which the minimax theorem holds

Extends spectral analysis to non-negative matrices with hourglass property

Abstract

We prove the minimax equality for the spectral radius $\rho(AB)$ of the product of matrices $A\in\mathcal{A}$ and $B\in\mathcal{B}$, where $\mathcal{A}$ and $\mathcal{B}$ are compact sets of non-negative matrices of dimensions $N\times M$ and $M\times N$, respectively, satisfying the so-called hourglass alternative.