Asymptotic Results on the Spectra of Block Toeplitz Preconditioned Matrices

It is well known that the generating function f ∈ L([−π, π],R) of a class of Hermitian Toeplitz matrices An(f) describes very precisely the spectrum of each matrix of the class [U. Grenader and G. Szegö, Toeplitz Forms and Their Applications, 2nd ed., Chelsea, New York, 1984; E. E. Tyrtyshnikov, Linear Algebra Appl., 232 (1996), pp. 1–43]. In this paper we consider n×n block Toeplitz matrices with m×m blocks generated by a Hermitian matrix-valued generating function f ∈ L([−π, π],C) and, in particular, we analyze the associated problem of preconditioning. Using previous results on this topic [P. Tilli and M. Miranda, SIAM J. Matrix Anal. Appl., to appear], we extend some theorems to this case that were proved in the one-level Toeplitz case [F. Di Benedetto, G. Fiorentino, and S. Serra, Comput. Math. Appl., 25 (1993), pp. 35–45; S. Serra, SIAM J. Matrix Anal. Appl., 17 (1996), pp. 1007–1019; S. Serra, Calcolo, 32 (1995), pp. 153–176] as well as in the two-level Toeplitz case [S. Serra, Linear Algebra Appl., 270 (1998), pp. 109–129]. This idea seems promising when dealing with linear systems arising from control theory and Markov chains theory [R. Preuss, Workshop on Toeplitz matrices in Filtering and Control, Santa Barbara, CA, August 1996; M. Neuts, Structured Stochastic Matrices of M/G/1 Type and Their Applications, Dekker, New York, 1989; E. Cinlar, Introduction to Stochastic Processes, Prentice–Hall, Englewood Cliffs, NJ, 1975].