A geometric approach to calculating the limit set of eigenvalues for banded Toeplitz matrices
(2023) In Master's Theses in Mathematical Sciences FMAM05 20222Mathematics (Faculty of Engineering)
- Abstract
- This thesis is about the limiting eigenvalue distribution of n × n Toeplitz matrices as n → ∞. The two classical questions we want to answer are: what is the limit set of the eigenvalues, and what is the limiting distribution of the eigenvalues. Our main result is a new approach to calculate the limit set Λ(b) for a Laurent polynomial b, i.e. for banded Toeplitz matrices. The approach is geometrical and based on the formula Λ(b) = ∩ᵨ sp T(bᵨ) with ρ ∈ (0, ∞). We show that the full intersection can be approximated by the intersection for a finite number of ρ's, and that sp T(bᵨ) can be well approximated by a polygon. This results in an algorithm whose output we show converge to Λ(b) in the Hausdorff metric. We implement the algorithm in... (More)
- This thesis is about the limiting eigenvalue distribution of n × n Toeplitz matrices as n → ∞. The two classical questions we want to answer are: what is the limit set of the eigenvalues, and what is the limiting distribution of the eigenvalues. Our main result is a new approach to calculate the limit set Λ(b) for a Laurent polynomial b, i.e. for banded Toeplitz matrices. The approach is geometrical and based on the formula Λ(b) = ∩ᵨ sp T(bᵨ) with ρ ∈ (0, ∞). We show that the full intersection can be approximated by the intersection for a finite number of ρ's, and that sp T(bᵨ) can be well approximated by a polygon. This results in an algorithm whose output we show converge to Λ(b) in the Hausdorff metric. We implement the algorithm in python and test it. It performs on par to and better in some cases than existing algorithms. We argue, but do not prove, that the average time complexity of the algorithm is O(n² + mn), where n is the number of ρ's and m is the number of vertices for the polygons approximating sp T(bᵨ). Further, we present the theory for Toeplitz matrices for symbols in the Wiener algebra, with an emphasis on the limiting eigenvalue distribution. In particular, we derive the limiting measure and limit set for hermitian Toeplitz matrices and banded Toeplitz matrices. (Less)
- Popular Abstract
- Mathematics can be used to model almost everything. A crucial concept to understanding mathematical models are eigenvalues. If we understand how the eigenvalues behave, we understand stability of a system, and how a system responds to changes. This work presents a geometric tool for understanding eigenvalues.
Eigenvalues can be interpreted as how much a system changes when the input is varied in a certain way. This work investigates how the eigenvalues of specific type of model behaves when the dimensions of the system grow very large. An eigenvalue is just a number, and there are as many eigenvalues for a system as there are dimensions. When the number of dimensions for the system grows to around one billion — which it does for some... (More) - Mathematics can be used to model almost everything. A crucial concept to understanding mathematical models are eigenvalues. If we understand how the eigenvalues behave, we understand stability of a system, and how a system responds to changes. This work presents a geometric tool for understanding eigenvalues.
Eigenvalues can be interpreted as how much a system changes when the input is varied in a certain way. This work investigates how the eigenvalues of specific type of model behaves when the dimensions of the system grow very large. An eigenvalue is just a number, and there are as many eigenvalues for a system as there are dimensions. When the number of dimensions for the system grows to around one billion — which it does for some models in statistical physics — there are a lot of eigenvalues to keep track of.
One can think of the eigenvalues as points in the plane, and as we add a new dimension to the system a new point appear, and all old points move a little. Instead of keeping track of all the points individually, we want to understand if and where they tend to move as the number of dimensions grow, or more technically, where they accumulate.
The main result of the thesis is a tool for calculating where the eigenvalues accumulate, i.e. the limit set for a special type of model. Previous results in the field show that the limit set can be described as the common area of an infinite number of different areas in the plane. But to handle an infinite number of areas on a computer is impossible. However, we show that only a finite number of those areas are needed to get a good approximation of the limit set. Also, we show that it is computationally possible to get a good approximation of the common area of the finite set of different areas. All in all, we present an algorithm that we implement, which is capable of producing a good approximation of the limit set for practical examples. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/student-papers/record/9109060
- author
- Bucht, Teodor LU
- supervisor
- organization
- course
- FMAM05 20222
- year
- 2023
- type
- H2 - Master's Degree (Two Years)
- subject
- keywords
- Toeplitz matrices, Banded Toeplitz matrices, Limit set, Eigenvalues
- publication/series
- Master's Theses in Mathematical Sciences
- report number
- LUTFMA-3493-2023
- other publication id
- 2023:E1
- language
- English
- additional info
- Uppladdningen godkänd av expedition@math.lth.se
- id
- 9109060
- date added to LUP
- 2023-02-27 09:42:33
- date last changed
- 2023-02-27 09:42:33
@misc{9109060,
abstract = {{This thesis is about the limiting eigenvalue distribution of n × n Toeplitz matrices as n → ∞. The two classical questions we want to answer are: what is the limit set of the eigenvalues, and what is the limiting distribution of the eigenvalues. Our main result is a new approach to calculate the limit set Λ(b) for a Laurent polynomial b, i.e. for banded Toeplitz matrices. The approach is geometrical and based on the formula Λ(b) = ∩ᵨ sp T(bᵨ) with ρ ∈ (0, ∞). We show that the full intersection can be approximated by the intersection for a finite number of ρ's, and that sp T(bᵨ) can be well approximated by a polygon. This results in an algorithm whose output we show converge to Λ(b) in the Hausdorff metric. We implement the algorithm in python and test it. It performs on par to and better in some cases than existing algorithms. We argue, but do not prove, that the average time complexity of the algorithm is O(n² + mn), where n is the number of ρ's and m is the number of vertices for the polygons approximating sp T(bᵨ). Further, we present the theory for Toeplitz matrices for symbols in the Wiener algebra, with an emphasis on the limiting eigenvalue distribution. In particular, we derive the limiting measure and limit set for hermitian Toeplitz matrices and banded Toeplitz matrices.}},
author = {{Bucht, Teodor}},
language = {{eng}},
note = {{Student Paper}},
series = {{Master's Theses in Mathematical Sciences}},
title = {{A geometric approach to calculating the limit set of eigenvalues for banded Toeplitz matrices}},
year = {{2023}},
}