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An Investigation Into Crouzeix's Conjecture

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 supz∈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.

Identiferoai:union.ndltd.org:CALPOLY/oai:digitalcommons.calpoly.edu:theses-4051
Date01 June 2022
CreatorsRoyston, Timothy T
PublisherDigitalCommons@CalPoly
Source SetsCalifornia Polytechnic State University
Detected LanguageEnglish
Typetext
Formatapplication/pdf
SourceMaster's Theses

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