Spektar matrice

master's thesis

• Title (english): Spectrum of matrices
• Author: Ante Gakić
• Mentor: Ljiljana Arambašić (mentor)
• Committee member: Ljiljana Arambašić (predsjednik povjerenstva)
• Committee member: Goran Igaly (član povjerenstva)
• Committee member: Veronika Pedić Tomić (član povjerenstva)
• Committee member: Matej Mihelčić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2024-07-19, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Ovaj diplomski rad istražuje spektar matrica, počevši od temeljnih pojmova za linearne operatore na konačnodimenzionalnim vektorskim prostorima i analognih tvrdnji za matrice. Zatim se istražuju metode računanja svojstvenih vrijednosti, uključujući razne oblike metode potencija i teorem o Geršgorinovim krugovima koji određuje položaj svojstvenih vrijednosti u kompleksnoj ravnini. Konačno, prikazuju se praktične primjene svojstvenih vrijednosti na pojednostavljenim modelima Markovljevih lanaca, matričnog modela rasta populacije, sustava rangiranja sportskih timova te Googleovog algoritma za sortiranje rezultata pretraživanja.

Abstract (english)

This thesis delves into the spectrum of matrices, starting with fundamental concepts for linear operators on finitedimensional vector spaces and analogous statements for matrices. It then investigates methods for calculating eigenvalues including various forms of the power method and Gerschgorin's Disk Theorem that locates the eigenvalues in the complex plane. The thesis concludes by presenting the practical applications of eigenvalues in simplified models of Markov chains, matrix population growth model, sports team ranking systems, and Google's PageRank algorithm for organizing search results.

• Keywords: linearni operatori; teorem o Geršgorinovim krugovima; Markovljevi lanaci
• Language: croatian
• URN:NBN: urn:nbn:hr:217:722570
• Study programme: Title: Mathematics Education; specializations in: Mathematics Education; Course: Mathematics Education; Study programme type: university; Study level: graduate; Academic / professional title: sveučilišni magistar edukacije matematike (sveučilišni magistar edukacije matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2024-07-24 11:23:15