Perron-Frobenius theorem for nonnegative multilinear forms and extensions

TL;DR

This paper extends the Perron-Frobenius theorem to nonnegative multilinear forms and polynomial maps, establishing conditions for a unique eigenvector and analyzing the convergence rate of the power algorithm.

Contribution

It provides a novel generalization of Perron-Frobenius theorem to multilinear and polynomial settings with nonnegative coefficients.

Findings

Proves Perron-Frobenius type theorem for multilinear forms.

Determines geometric convergence rate of the power algorithm.

Ensures uniqueness of the normalized eigenvector.

Abstract

We prove an analog of Perron-Frobenius theorem for multilinear forms with nonnegative coefficients, and more generally, for polynomial maps with nonnegative coefficients. We determine the geometric convergence rate of the power algorithm to the unique normalized eigenvector.