Inside the eigenvalues of certain Hermitian Toeplitz band matrices

Abstract

MSC:15A18 41A25 47B35 65F15 Keywords:Toeplitz matrix Eigenvalue Asymptotic expansions a b s t r a c t While extreme eigenvalues of large Hermitian Toeplitz matrices have been studied in detail for a long time, much less is known about individual inner eigenvalues. This paper explores the behavior of the jth eigenvalue of an n-by-n banded Hermitian Toeplitz matrix as n tends to infinity and provides asymptotic formulas that are uniform in j for 1 ≤ j ≤ n. The realvalued generating function of the matrices is assumed to increase strictly from its minimum to its maximum, and then to decrease strictly back from the maximum to the minimum, having nonzero second derivatives at the minimum and the maximum. The results, which are of interest in numerical analysis, probability theory, or statistical physics, for example, are illustrated and underpinned by numerical examples.

Asymptotics of Individual Eigenvalues of a Class of Large Hessenberg Toeplitz Matrices

“…If a is real-valued, then the matrices T n (a) are all Hermitian, and in this case a number of results on the asymptotics of the eigenvalues of T n (a) is known; see, for example, [6], [7], [13], [16], [18], [20], [21], [23], [24], [26], [27], [29], [30]. We here consider genuinely complex-valued symbols, in which case the overall picture is less complete.…”

“…The asymptotic behavior of the extreme eigenvalues of Hermitian Toeplitz matrices is fairly well understood; see the references cited above. Paper [6] contains asymptotic expansions for individual inner eigenvalues of certain banded Hermitian Toeplitz matrices. The recent papers [11] and [17] concern asymptotic formulas for individual eigenvalues of Toeplitz matrices whose symbols are complex-valued and have a so-called Fisher-Hartwig singularity.…”

Weighted angle Radon transform: Convergence rates and efficient estimation

“…In particular, the conditiond k = d −k ensures that λ is real. Using the results of Böttcher, Grudsky, and Maksimenko (2010) on the uniform behavior of the eigenvalues of banded Toeplitz matrices A m , we obtain Proposition 2. Suppose λ is banded and satisfies λ(−π/2) = λ(π/2) = 0.…”

A matrix-less and parallel interpolation–extrapolation algorithm for computing the eigenvalues of preconditioned banded symmetric Toeplitz matrices

“…In the case where u = 1 identically and v satisfies some additional assumptions, Conjecture 1 was formally proved by Bogoya, Böttcher, Grudsky, and Maximenko in a sequence of recent papers [6,8,10]. Assuming Conjecture 1, in Section 2 of this paper, we describe and analyze a new algorithm for computing the eigenvalues of X n = T n (u) −1 T n (v); and in Section 3, we illustrate its performance through numerical experiments.…”