Asymptotic Spectra of Hermitian Block Toeplitz Matrices and Preconditioning Results

We study the asymptotic behavior of the eigenvalues of Hermitian n × n block Toeplitz matrices T n , with k × k blocks, as n tends to infinity. No hypothesis is made concerning the structure of the blocks. Such matrices {T n } are generated by the Fourier coefficients of a Hermitian matrix-valued function f∈ L², and we study the distribution of their eigenvalues for large n, relating their behavior to some properties of f as a function; in particular, we show that the distribution of the eigenvalues converges to a limit μf, and we explicitly compute μf in terms of f, showing that ∫ F\, dμf=1/k∫ F(f). Some consequences of this distribution and some localization results for the eigenvalues of T n are discussed. We also study the eigenvalues of the preconditioned matrices {P n -1 T n }, where the sequence {P n } is generated by a positive definite matrix-valued function p. We show that the spectrum of any P n -1 T n is contained in the interval [r,R], where r is the smallest and R the largest eigenvalue of p -1 f . We also prove that the first m eigenvalues of P n -1 T n tend to r and the last m tend to R, for any fixed m. Finally, the exact limit value of the condition number of the preconditioned matrices is computed.