Eigenvalues of Locally Perturbed Toeplitz Matrices

 
Toeplitz matrices enjoy the dual virtues of beauty and ubiquity.
We begin this talk by surveying some of the interesting spectral
properties of such matrices, emphasizing the distinctions between
infinite-dimensional Toeplitz matrices and the large-dimensional
limit of the corresponding finite matrices.
 
These basic results utilize the algebraic Toeplitz structure,
but in many applications, one is forced to spoil this structure
with some perturbations (e.g., by imposing boundary conditions
upon a finite difference discretization of an initial-boundary
value problem).  How do such perturbations affect the eigenvalues?
 
This talk will address this question for "localized" perturbations,
i.e., perturbations that are restricted to a single entry, or a block
of entries whose size remains fixed as the matrix dimension grows.
One can show, for a broad class of matrices, that sufficiently small
perturbations fail to alter the spectrum, though the spectrum is
exponentially sensitive to other kinds of perturbations.
For larger real single-entry perturbations, one observes the perturbed
eigenvalues trace out curves in the complex plane.  We'll show a number
of illustrations of this phenomenon for tridiagonal Toeplitz matrices,
which arise in many applications.
 
This talk describes collaborative work with Albrecht Boettcher,
Marko Lindner, and Viatcheslav Sokolov of TU Chemnitz.