Background

A Toeplitz matrix is an \(N\times N\) matrix \(A_N\) (with \(N\in\mathbb N\) or \(N=\infty\)) with constant entries along the diagonals, \[A_N=\begin{pmatrix} a_0& a_{-1} & a_{-2}& \\ a_{1} & a_0 & a_{-1} & \ddots \\ a_{2}& a_{1} & a_0 & \ddots \\ & \ddots & \ddots & \ddots\end{pmatrix},\] where \(a_n\in\mathbb C\) for \(n\in\mathbb Z\). This means that the \((i,j)\) entry of \(A_N\) is \(a_{i-j}\). Toeplitz matrices are closely connected with the Fourier series \[f(\varphi)=\sum_{n\in\mathbb Z}a_n{\rm e}^{{\rm i}n\varphi}.\] We are interested in the spectrum of \(A_N\), which is defined as \[\sigma(A_N)=\{z\in\mathbb C:\,A_N-z I \,\,\text{not bijective}\}.\] For a finite matrix this is the set of eigenvalues, whereas for an infinite matrix the spectrum may contain additional points. The spectrum consists of the points where \(z\mapsto \|(A_N-zI)^{-1}\|\) has a singularity (for an appropriate Matrix norm).

Here you see an example of a Toeplitz matrix. The eigenvalues of \(A_N\) with \(N=250\) are in blue colour. In red colour is the Fourier series curve \(S=\{f(\varphi):\,\varphi\in [0,2\pi)\}\) (which is called symbol curve). The spectrum of the corresponding infinite matrix \(A_{\infty}\) is the symbol curve together with some of the connected components it encloses (namely the components where the winding number of the symbol curve is non-zero).

In general, not every \(z\in\sigma(A_{\infty})\) is the limit of a sequence \((z_N)_{N\in\mathbb N}\) of \(z_N\in\sigma(A_N)\). However, it turns out that for every \(z\in\sigma(A_{\infty})\) we have \(\|(A_N-zI)^{-1}\|\to\infty\) as \(N\to\infty\). This motivates to study also the behaviour of the set (called \(\varepsilon\)-pseudospectrum) \[\sigma_{\varepsilon}(A_N)=\{z\in\mathbb C:\,\|(A_N-zI)^{-1}\|>1/\varepsilon\}\] as \(N\to\infty\) (for \(\varepsilon>0\) small).

Description of the project

You will familiarise yourself with the concept of Toeplitz matrices. Using Chapter 1 of reference [2] you will learn about the basic properties of \(A_N\) and its spectrum \(\sigma(A_N)\). The book is a good introduction into the topic and contains also helpful exercises. Then, individually, you will focus on different aspects, e.g. the behaviour of \(\sigma(A_N)\) or \(\sigma_{\varepsilon}(A_N)\) as \(N\to\infty\) (relevant here are Chapters 7,11,12 of [2], and [3] is an excellent introduction for pseudospectrum). This can be done from a purely theoretical or numerical point of view.

Prerequisites

None. If you are interested in numerical implementations, it might be helpful to have some prior knowledge in any programming language/software. It is suggested (but not necessary) to take Functional Analysis and Applications IV at the same time as the project.

References

[1] https://en.m.wikipedia.org/wiki/Toeplitz_matrix

[2] A.Böttcher, S.Grudsky. Spectral Properties of Banded Toeplitz Matrices. SIAM (2005). Online access via Durham library.

[3] L.N. Trefethen and M. Embree. Spectra and pseudospectra: the behavior of nonnormal matrices and operators. Princeton University Press (2005). Online access via Durham library.