Extremal eigenvalues of infinite tridiagonal matrices

Abstract

Although tridiagonal matrices have many applications, such as in linear models where objects are influenced only by their neighbouring objects, few methods exist to compute their eigenvalues analytically. The present thesis introduces a few theorems and ideas in order to do so. Its main goal is to find analytical approximations of the eigenvalues of tridiagonal matrices, in particular to provide statements on the behaviour of the eigenvalues as the size of the matrix grows to infinity.

Two classes of matrices are treated. For tridiagonal Toeplitz matrices we use the connection between tridiagonal matrices and second order recurrence relations, in order to find an explicit formula for the eigenvalues. For general matrices an estimate on the absolutely smallest negative eigenvalue is given, which always functions as a lower bound when dealing with negative definite matrices. These bounds depend on the trailing coefficients of the characteristic polynomial. We provide a method to compute these trailing coefficients through two recurrence relations and show there to be situations where we can estimate the smallest eigenvalue through this method, without being able to calculate it explicitly.

The determined upper and lower bound are also applied to the SIS model, a well-known compartmental epidemiological model. They are used to provide statements on the behaviour of the absolutely smallest eigenvalue in a linear finite state variant of the model. In particular we find that this eigenvalue will converge to 0 exponentially if the infection rate is larger than the recovery rate. In other situations the convergence to 0 is either proven to be of a way lower rate, or non-existent.