Asymptotics of the smallest singular value of a class of toeplitz–generated matrices and related finite rank perturbations

Abstract

Square matrices of the form Xn = Tn+fn(T −1 n ) ∗ , where Tn is an n×n invertible banded Toeplitz matrix and fn some positive sequence are considered. The norms of their inverses are described asymptotically as their size n increases. Certain finite rank perturbations of these matrices are shown to have no effect on this behaviour.