On large Toeplitz band matrices with an uncertain block Original Research Article

This paper investigates the possible spectra of large, finite dimensional Toeplitz band matrices with perturbations (impurities, uncertainties) in the upper-left m×m block. The main result shows that the asymptotic spectrum of such a matrix is not affected by these perturbations, provided they have sufficiently small norm. This follows from analysis of structured pseudospectra (structured spectral value sets). In contrast, for typical non-Hermitian Toeplitz matrices there exist certain rank-one perturbations of arbitrarily small norm that move an eigenvalue away from the asymptotic spectrum in the large-dimensional limit.