An Investigation Into Crouzeix's Conjecture

Author

Timothy T. Royston, California Polytechnic State University, San Luis ObispoFollow

DOI: https://doi.org/10.15368/theses.2022.42
Available at: https://digitalcommons.calpoly.edu/theses/2460

Date of Award

6-2022

Degree Name

MS in Mathematics

Department/Program

Mathematics

College

College of Science and Mathematics

Advisor

Linda Patton

Advisor Department

Mathematics

Advisor College

College of Science and Mathematics

Abstract

We will explore Crouzeix’s Conjecture, an upper bound on the norm of a matrix after the application of a polynomial involving the numerical range. More formally, Crouzeix’s Conjecture states that for any n × n matrix A and any polynomial p from C → C,
∥p(A)∥ ≤ 2 sup_{z∈W (A)} |p(z)|.
Where W (A) is a set in C related to A, and ∥·∥ is the matrix norm. We first discuss the conjecture, and prove the simple case when the matrix is normal. We then explore a proof for a class of matrices given by Daeshik Choi. We expand upon the proof where details are left out in the original. We also find and fix a small flaw in one section of the original paper.