Spectra of Tridiagonal Matrices

TL;DR

This paper characterizes the eigenvalues and eigenvectors of complex tridiagonal matrices with arbitrary boundary conditions, revealing how boundary choices influence spectral properties and system dynamics.

Contribution

It provides explicit asymptotic descriptions of both regular and special eigenvalues and eigenvectors for large matrices with arbitrary boundary conditions.

Findings

Regular eigenvalues vary little with boundary conditions

Special eigenvalues depend sensitively on boundary conditions

Eigenvectors are determined up to order 1/n

Abstract

We characterize the eigenvalues and eigenvectors of a class of complex valued tridiagonal $n$ by $n$ matrices subject to arbitrary boundary conditions, i.e. with arbitrary elements on the first and last rows of the matrix. %By boundary conditions, we mean the first and last row of the matrix. For large $n$, we show there are up to $4$ eigenvalues, the so-called \emph{special eigenvalues}, whose behavior depends sensitively on the boundary conditions. The other eigenvalues, the so-called \emph{regular eigenvalues} vary very little as function of the boundary conditions. For large $n$, we determine the regular eigenvalues up to ${\cal O}(n^{-2})$, and the special eigenvalues up to ${\cal O}(\kappa^n)$, for some $\kappa\in (0,1)$. The components of the eigenvectors are determined up to ${\cal O}(n^{-1})$. The matrices we study have important applications throughout the sciences. Among the most common ones are arrays of linear dynamical systems with nearest neighbor coupling, and discretizations of second order linear partial differential equations. In both cases, we give examples where specific choices of boundary conditions substantially influence leading eigenvalues, and therefore the global dynamics of the system.