A geometric approach to calculating the limit set of eigenvalues for banded Toeplitz matrices

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.