Extreme singular values and eigenvalues of non-Hermitian block Toeplitz matrices

摘要

In this paper we are concerned with the analysis of the distribution and localization of the singular values of Toeplitz matrices {Tn(f)} generated by a p-variate Lebesgue integrable matrix-valued function f:Qp→Ch×k,Q=(−π,π). We prove that the union of the essential ranges of the singular values of f is a proper/weak cluster for the whole set of the singular values of {Tn(f)}, by showing that the number of outliers is strongly depending on the regularity features of the underlying function f: in particular, if f is continuous or from the Krein algebra and p=1, then the cluster is proper. Other results concerning the extreme spectral behavior of {Tn(f)}, second-order ergodic formulas and localization of eigenvalues of preconditioned matrices {Tn−1(g)Tn(f)} are presented. Some examples of applications to the preconditioning of these results are also discussed.