A p-adic Perron-Frobenius Theorem

TL;DR

This paper establishes a p-adic analogue of the Perron-Frobenius theorem, showing that certain matrices over p-adic fields have a dominant eigenvalue and their normalized powers converge to a projection onto its eigenspace.

Contribution

It proves a new Perron-Frobenius type theorem for matrices over p-adic fields, extending classical results to non-Archimedean settings.

Findings

Existence of a strictly maximal eigenvalue for matrices over ${f Q}_p$

Convergence of normalized matrix powers to a projection operator

Extension of Perron-Frobenius theory to non-Archimedean fields

Abstract

We prove that if an $n\times n$ matrix defined over ${\mathbb Q}_p$ (or more generally an arbitrary complete, discretely-valued, non-Archimedean field) satisfies a certain congruence property, then it has a strictly maximal eigenvalue in ${\mathbb Q}_p$, and that iteration of the (normalized) matrix converges to a projection operator onto the corresponding eigenspace. This result may be viewed as a $p$-adic analogue of the Perron-Frobenius theorem for positive real matrices.