Dynamical spectrum via determinant-free linear algebra

TL;DR

This paper introduces a determinant-free linear algebra approach to estimate eigenvalues of matrices linked to Markov dynamical systems, enabling simultaneous analysis of mixing rates, symmetries, and spectral properties without direct matrix calculations.

Contribution

It presents a novel determinant-free linear algebra method for analyzing eigenvalues and spectral properties of matrices in Markov dynamical systems, allowing simultaneous estimates.

Findings

Eigenvalues estimated without matrix calculations

Mixing rates for all systems obtained simultaneously

Spectral properties of related factor systems characterized

Abstract

We consider a sequence of matrices that are associated to Markov dynamical systems and use determinant-free linear algebra techniques (as well as some algebra and complex analysis) to rigorously estimate the eigenvalues of every matrix simultaneously without doing any calculations on the matrices themselves. As a corollary, we obtain mixing rates for every system at once, as well as symmetry properties of densities associated to the system; we also find the spectral properties of a sequence of related factor systems.