Block Toeplitz matrices and preconditioning

Abstract

We study the asymptotic behaviour of the eigenvalues of Hermitiann×n block Topelitz matricesT _n , withk×k blocks, asn 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 functionf∈L ², and we study the distribution of their eigenvalues for largen, relating their behaviour to some properties of the functionf. We also study the eigenvalues of the preconditioned matrices{P ⁻¹_n T_n}, where the sequence{P _n } is generated by a positive definite matrix valued functionp. We show that the spectrum of anyP ⁻¹ _n T _n is contained in the interval [r, R], wherer is the smallest andR the largest eigenvalue ofp ⁻¹ f. We also prove that the firstm eigenvalues ofP ⁻¹_n T_n tend tor and the lastm tend toR, for anym fixed. Finally, exact limit values for both the condition number and the conjugate gradient convergence factor for the preconditioned matricesP ⁻¹_n T_n are computed.