Toeplitz matrices enjoy the dual virtues of ubiquity and beauty. 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, by which we mean perturbations that are restricted to a single entry, or a block of entries whose size remains fixed as the matrix dimension grows. One finds, for a broad class of matrices, that sufficiently small perturbations fail to alter the spectrum, though the spectrum is exponentially sensitive to other 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.
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This talk describes collaborative work with Albrecht Boettcher, Marko Lindner, and Viatcheslav Sokolov of TU Chemnitz.