Normalne matrice

master's thesis

• Author: Ana Šimić
• Mentor: Rajna Rajić (mentor)
• Committee member: Rajna Rajić (predsjednik povjerenstva)
• Committee member: Damir Bakić (član povjerenstva)
• Committee member: Zrinka Franušić (član povjerenstva)
• Committee member: Vjeran Hari (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2017-09-25, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Tema ovog rada su normalne matrice. U prvom poglavlju rada prikazujemo osnovne rezultate o normalnim matricama. Dajemo neke karakterizacije normalnih matrica. Razmatramo (0,1)-matrice te dajemo dovoljan uvjet za njihovu normalnost. Također razmatramo nejednakosti koje uključuju elemente matrica te svojstvene i singularne vrijednosti matrica. Slučajevi jednakosti u tim nejednakostima rezultiraju normalnošću matrica. Neki rezultati o normalnim matričnim pertubacijama su također prikazani. Drugo poglavlje usmjereno je na podklasu normalnih matrica, unitarne matrice. Prikazujemo osnovna svojstva unitarnih matrica, raspravljamo o strukturi realnih ortogonalnih matrica s obzirom na transformacije sličnosti, povezujemo kontrakcije s unitarnim matricama, te prezentiramo nejednakost koja povezuje prosječne svojstvene vrijednosti svake od dviju unitarnih matrica s prosječnom svojstvenom vrijednosti njihovog umnoška.

Abstract (english)

The subject of this work is normal matrices. In the first chapter of this work we present basic results on normal matrices. We give some characterizations of a normal matrix. We consider (0; 1)-matrices, and give a sufficient condition on their normality. We also consider the inequalities involving entries, eigenvalues and singular values of matrices. The equality cases of these inequalities result in the normality of the matrices. Some results on normal matrix pertubations are presented as well. The second chapter focuses on a subclass of normal matrices, unitary matrices. We present basic properties of unitary matrices, discuss the structure of real orthogonal matrices under similarity, deal with the connections of contractions with unitary matrices, and present an inequality relating the average of the eigenvalues of each of two unitary matrices to that of their product.

• Keywords: normalne matrice; unitarne matrice; realne ortogonalne matrice
• Keywords (english): normal matrices; unitary matrices; real orthogonal matrices
• Language: croatian
• URN:NBN: urn:nbn:hr:217:325049
• 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: 2018-03-22 09:47:22