This project aims to introduce techniques of matrix analysis and apply them both rigorously and numerically to study the pseudospectra of Toeplitz and related (block) matrices. Of particular interest is the structured pseudospectra which appear in many applications to computer science.
Department: Mathematics & Statistics
Supervised by: Jani Virtanen
In perturbation analysis it is natural to ask what happens to the spectrum of a matrix when it is perturbed by matrices of the same structure. While pseudospectra are well-studied and vast literature exists on the subject (see [TE] below and the references therein), much less is known about structured pseudospectra of matrices, especially of matrices that possess block structures. Motivation for structured pseudospectra comes from applications, such as floating-point error analysis; situations where the entries are affected by experimental uncertainty; backward error analysis, numerical algorithms and other spectral problems in linear algebra; problems in control theory; and stability theory for dynamical systems When the entries of the matrix are numbers, it is known that the structured and unstructured pseudospectra equal for many classes of matrices. The purpose of this project is to study similar questions for classes of block matrices whose entries are matrices instead of numbers. This problem was recently solved in special cases (see [FV]) and some further progress was recently made in [YHC], which we aim to fully understand and extend to other structures. References: [FV] Ferro, R., Virtanen, J. A.: A note on structured pseudospectra of block matrices. J. Comput. Appl. Math. (2017). [TE] Trefethen, L. N., Embree, M.: Spectra and pseudospectra: The behavior of nonnormal matrices and operators. Princeton University Press, 2005. [YHC] Yu, Y., Hou, G., Chen, A.: Comments on "A note on structured pseudospectra of block matrices'' [J. Comput. Appl. Math. 322 (2017)]. J. Comput. Appl. Math. (2020).
In the first two weeks the student will learn the basic results on pseudospectra and their applications. This requires knowledge of linear algebra and basic mathematical analysis. The following two weeks will be used to study the pseudospectra of Toeplitz and related matrices and start using mathematical software (such as MatLab or Julia) to numerically investigate the properties of such matrices. The last two weeks will be used to study and extend a recent research article that relates to PI’s work in the field. The approach will depend on the student's background and preferences, and in particular it will involve more rigorous analysis or be computational/software based.
The student should have good knowledge of Linear Algebra, Real and Complex Analysis, Algebra I, and have some skills in the use of MatLab and/or other mathematical software or programming language (such as Julia). The use of LaTeX, which is widely used in academia for the communication and publication of scientific documents in many fields, would be a plus but not expected and the necessary skills will be taught during the project.
The student will engage in pure mathematics research in an area that is currently of high interest. There will also be opportunities to develop some skills and methods to use computer software (such as MatLab and Mathematica) to study some computational aspects of the matrix analysis. When a problem is formulated in pure mathematics, there are no guarantees that it will lead to a publication unless the problem has been previously solved. However, the problem is chosen in such a way that it is likely that the placement may well lead to a publication in an international journal in pure mathematics. Also, the student will be able to contribute to and observe all stages of the research to be carried out as an active member of the team.
Department of Mathematics with meetings in PI’s office and classrooms in the Mathematics Building, or online when necessary.
Normally 9 to 5 but can be flexible.
Monday 14 June 2021 - Friday 23 July 2021
This project will be advertised until 5pm on Friday 14th May. Students should submit their CV and Cover Letter directly to the Project Supervisor (click on supervisor name at the top of the page for email). Successful candidates will be invited for an interview.