Tridiagonal Matrices and Boundary Conditions

TL;DR

This paper analyzes the spectra of tridiagonal matrices from differential equations modeling flocking, showing how boundary conditions influence the stability of consensus and flocking systems.

Contribution

It provides a spectral analysis of tridiagonal matrices with arbitrary boundary conditions, linking boundary choices to system stability in flocking models.

Findings

Spectral properties depend on boundary conditions.

Asymptotic stability varies with boundary choices.

Results inform design of stable flocking systems.

Abstract

We describe the spectra of certain tridiagonal matrices arising from differential equations commonly used for modeling flocking behavior. In particular we consider systems resulting from allowing an arbitrary boundary condition for the end of a one dimensional flock. We apply our results to demonstrate how asymptotic stability for consensus and flocking systems depends on the imposed boundary condition.