Spectral and computational analysis of block Toeplitz matrices having nonnegative definite matrix-valued generating functions

It is well known that the generating function f ∈ L1([-π, π], ℜ) of a class of Hermitian Toeplitz matrices {An(f)}n describes very precisely the spectrum of each matrix of the class. In this paper we consider n x n Hermitian block Toeplitz matrices with m x m blocks generated by a Hermitian matrix-valued generating function f ∈ L1([-π, π], Cm x m). We extend to this case some classical results by Grenander and Szegö holding when m = 1 and we generalize the Toeplitz preconditioning technique introduced in the scalar case by R. H. Chan and F. Di Benedetto, G. Fiorentino and S. Serra. Finally, concerning the spectra of the preconditioned matrices, some asymptotic distribution properties are demonstrated and, in particular, a Szegö-style theorem is proved. A few numerical experiments performed at the end of the paper confirm the correctness of the theoretical analysis.