Geometrijske primjene numeričke slike matrice

master's thesis

• Title (english): Geometric applications of the numerical matrix image
• Author: Ivana Šanje
• Mentor: Juraj Šiftar (mentor)
• Committee member: Juraj Šiftar (predsjednik povjerenstva)
• Committee member: Ana Prlić (član povjerenstva)
• Committee member: Aleksandra Čižmešija (član povjerenstva)
• Committee member: Sanja Varošanec (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2022-07-20, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom diplomskom radu obrađena je tema geometrijskih primjena numeričke slike operatora, koja se specijalno bavi kompleksnim matricama reda 2. U prvom poglavlju je definirana numerička slika, te su dana neka od njenih svojstava koja su ključna u geometrijskim dokazima. Zatim, u sljedećem poglavlju je detaljno opisano svojstvo konveksnosti numeričke slike matrice potkrijepljeno Teoremom o eliptičkoj slici matrice. U radu je prikazano i kako Teorem o eliptičkoj slici matrice omogućuje zajednički pristup dokazima teorema koji slijede, od kojih svaki na svoj način uključuje pojam elipse. U trećem poglavlju je dano nekoliko verzija Siebeckovog teorema gdje se opisuje veza između kompleksnog polinoma stupnja 3 i njegove derivacije. Nadalje, četvrto poglavlje prikazuje vezu Blaschkeova produkta iz kompleksne analize i Ponceletova teorema iz područja projektivne geometrije. Ovdje je fokus na Blaschkeovoj 3-elipsi i Ponceletovoj 3-elipsi koje su u svojoj suštini jednake. U konačnici je opisana međusobna povezanost ovih naizgled nepovezanih teorema.

Abstract (english)

The main topic of this master thesis are geometric applications of the numerical range of the operator, which specifically deals with the 2 × 2 complex matrices. The first chapter defines a numerical range and gives some of its properties that are crucial in geometric proofs. Then, in the next chapter, the convexity property of the numerical range of the matrix is described in detail, supported by the Elliptical Range Theorem. It is also shown that the Elliptical Range Theorem enables a common approach to the proofs of the following theorems and each of them in its own way includes the notion of an ellipse. In the third chapter several versions of Siebeck’s Theorem are shown, that describe the connection between a complex polynomial of degree 3 and its derivative. Furthermore, the fourth chapter presents the connection between Blaschke’s product from complex analysis and Poncelet’s Theorem from the field of projective geometry. Also, here is the focus on the Blaschke’s 3-ellipse and the Poncelet’s 3-ellipse which are essentially equal. Finally, connection between these theorems that seem unrelated at first glance will be given.

• Keywords: kompleksne matrice reda 2; svojstvo konveksnosti; Teorem o eliptičkoj slici matrice; Siebeckovo teorem; Blaschkeov produkt; Ponceletov teorem; Blaschkeova 3-elipsa; Ponceletova 3-elipsa
• Keywords (english): 2 × 2 complex matrices; convexity property; Elliptical Range Theorem; Siebeck’s Theorem; Blaschke’s product; Poncelet’s Theorem; Blaschke’s 3-ellipse; Poncelet’s 3-ellipse
• Language: croatian
• URN:NBN: urn:nbn:hr:217:684119
• Study programme: Title: Mathematics Education; specializations in: Mathematics Education; Course: Mathematics Education; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra edukacije matematike (magistar/magistra edukacije matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2022-09-14 12:12:29