The Perron-Frobenius theorem for multi-homogeneous mappings

TL;DR

This paper extends Perron-Frobenius theory to multi-homogeneous mappings, providing new spectral results, convergence analysis, and a unified framework for nonnegative matrices, tensors, and multilinear forms.

Contribution

It introduces the concept of order-preserving multi-homogeneous mappings, unifying existing theories and establishing new spectral properties and convergence results for these mappings.

Findings

Established Perron-Frobenius type results for multi-homogeneous mappings.

Proved a Collatz-Wielandt principle for the spectral radius.

Analyzed convergence of iterates to positive eigenvectors.

Abstract

The Perron-Frobenius theory for nonnegative matrices has been generalized to order-preserving homogeneous mappings on a cone and more recently to nonnegative multilinear forms. We unify both approaches by introducing the concept of order-preserving multi-homogeneous mappings, their associated nonlinear spectral problems and spectral radii. We show several Perron-Frobenius type results for these mappings addressing existence, uniqueness and maximality of nonnegative and positive eigenpairs. We prove a Collatz-Wielandt principle and other characterizations of the spectral radius and analyze the convergence of iterates of these mappings towards their unique positive eigenvectors. On top of providing a new extension of the nonlinear Perron-Frobenius theory to the multi-dimensional case, our contribution poses the basis for several improvements and a deeper understanding of the current spectral theory for nonnegative tensors. In fact, in recent years, important results have been obtained by recasting certain spectral equations for multilinear forms in terms of homogeneous maps, however as our approach is more adapted to such problems, these results can be further refined and improved by employing our new multi-homogeneous setting.