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.