Higher Rank Numerical Ranges for Certain Non-Normal Matrices

This thesis undertakes the study of higher rank numerical ranges for certain non-normal matrices, namely direct sums of Jordan blocks, and direct sums of the form (Jordan Block plus scalar multiple of the identity). The higher rank numerical range of a normal matrix is well-understood, but little is known in general, other than the fact that it is always compact and convex. Our main goal is to study the geometric behavior of the higher rank numerical ranges of certain non-normal matrices. In this thesis we show that the higher rank numerical ranges for the direct sums of Jordan blocks with the same eigenvalues behave as in the case of the n-dimensional shift matrix [Gaaya 2012], so it is a closed disk centred at the eigenvalue alpha. More importantly, for the case of direct sums of the form (Jordan block plus scalar multiple of the identity), we calculate the higher rank numerical range completely, and we set conditions which guarantee that the eigenvalues belong to the higher rank numerical range. Our results allow us to show explicit sharp examples of some extreme properties of higher-rank numerical ranges. We have also produced Javascript programs which draw higher rank numerical ranges; the first one draws the higher rank numerical ranges of direct sums of Jordan matrices and scalar multiples of the identity, while the second version will draw them for any square matrix---at the cost of losing precision as the size of the matrices grow larger (due to the growing complexity of calculating eigenvalues).

Name thesis:degree_name

Doctor of Philosophy (PhD)

Level thesis:degree_level

Doctoral -- first

Discipline thesis:degree_discipline

Mathematics

Grantor dc:publisher

Faculty of Graduate Studies and Research, University of Regina

Year dc:date.issued

2019

Author dc:creator

• Mustafa, Saleh Mohamed Ali

Advisor dc:contributor.advisor

• Argerami, Martin

Committee members dc:contributor.committeemember

• Floricel, Remus
• Szechtman, Fernando
• Zilles, Sandra

Language dc:language.iso

en

DOI

https://doi.org/10.82465/5151

OAI identifier oai:identifier

oai:uregina.scholaris.ca:10294/9176