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 Tn, with k times k blocks, as n tends to infinity. No hypothesis is made concerning the structure of the blocks. Such matrices {Tn} are generated by the Fourier coefficients of a Hermitian matrix-valued function f in L^2, 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 $\mu_f$, and we explicitly compute $\mu_f$ in terms of f, showing that $\int F\, d\mu_f=1/k\int\tr F(f)$. Some consequences of this distribution and some localization results for the eigenvalues of Tn are discussed. We also study the eigenvalues of the preconditioned matrices {Pn-1Tn}, where the sequence {Pn} is generated by a positive definite matrix-valued function p. We show that the spectrum of any Pn-1Tn is contained in the interval [r,R], where r is the smallest and R the largest eigenvalue of p-1f. We also prove that the first m eigenvalues of Pn-1Tn 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.