The First Order Asymptotics of the Extreme Eigenvectors of Certain Hermitian Toeplitz Matrices

Böttcher, Albrecht; Grudsky, Sergei M.; Maksimenko, E. A.; Unterberger, Jérémie

doi:10.1007/s00020-008-1646-x

Cited by 15 publications

(7 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…This work is also outlined on pages 256 to 259 of [7]. Recent work on and applications of extreme eigenvalues include the papers [8][9][10][11][12][13][14][15]. The purpose of this paper is to explore the behavior of λ (n) j inside the set of the eigenvalues, for example, the asymptotics of λ (n) j as n → ∞ and j/n → x ∈ (0, 1).…”

Section: Introduction and Main Resultsmentioning

confidence: 98%

See 1 more Smart Citation

Inside the eigenvalues of certain Hermitian Toeplitz band matrices

Böttcher¹,

Grudsky²,

Maksimenko³

2010

Journal of Computational and Applied Mathematics

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Introduction and Main Resultsmentioning

confidence: 98%

“…In this theorem, (6) ⇒ (7) ⇒ (8). Note that if the number j remains fixed, then O j 4 /n 4 = O 1/n 4 .…”

Section: Theorem 11mentioning

confidence: 86%

Inside the eigenvalues of certain Hermitian Toeplitz band matrices

Böttcher¹,

Grudsky²,

Maksimenko³

2010

Journal of Computational and Applied Mathematics

Self Cite

View full text Add to dashboard Cite

show abstract

“…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.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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

Bogoya

Böttcher

Grudsky

2012

Recent Progress in Operator Theory and Its Applications

Self Cite

View full text Add to dashboard Cite

Abstract. We study the asymptotic behavior of individual eigenvalues of the n-by-n truncations of certain infinite Hessenberg Toeplitz matrices as n goes to infinity. The generating function of the Toeplitz matrices is supposed to be of the form a(t) = t, where α is a positive real number but not an integer and f is a smooth function in H ∞ . The classes of generating functions considered here and in a recent paper by Dai, Geary, and Kadanoff are overlapping, and in the overlapping cases, our results imply in particular a rigorous justification of an asymptotic formula which was conjectured by Dai, Geary, and Kadanoff on the basis of numerical computations.MSC 2010. Primary 47B35. Secondary 15A15, 15A18, 47N50, 65F15

show abstract

“…For this problem, instead of a spatially invariant structure, employing the block-Toeplitz structure of system matrices seems promising. We are currently working on stability analysis based on the mathematical results of these structured matrices (Böttcher et al, 2009) …”

Section: Discussion On Extensionmentioning

confidence: 99%