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- W2058478885 abstract "We study the asymptotic behavior of the eigenvalues of Hermitian n × n block Toeplitz matrices Tn, with k × 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 $fin L^2$, 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 $int F, dmu_f=1/kinttr F(f)$. 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." @default.
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- W2058478885 date "2000-01-01" @default.
- W2058478885 modified "2023-09-27" @default.
- W2058478885 title "Asymptotic Spectra of Hermitian Block Toeplitz Matrices and Preconditioning Results" @default.
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- W2058478885 doi "https://doi.org/10.1137/s0895479896313036" @default.
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