Normalne matrike

delo diplomskega seminarja

Tim Mulej (Author), Roman Drnovšek (Mentor)

Abstract

Normalnost matrik je ena od bolj zanimivih poglavij linearne algebre. Ne samo zato, ker imajo normalne matrike razmeroma preprosto definicijo, ampak tudi zato, ker so uporabne v praksi, kar je razlog, da je bilo odkritih že $89$ karakterističnih lastnosti normalnih matrik. V tem delu smo si izbrali $25$ karakterističnih lastnosti in pokazali ekvivalence med njimi. Posvetili pa smo se tudi vprašanju, kako “blizu” sta si dve kvadratni matriki glede na njune lastne vrednosti. Ali še bolj zanimivo, kaj se zgodi z lastnimi vrednostmi matrike, če matriko malo perturbiramo. V tem delu smo na ti dve vprašanji odgovorili za normalne matrike.

Keywords

matrike;lastne vrednosti;lastni vektorji;Schurjev razcep;polarni razcep;Hoffman-Wielandtov izrek;Sunov izrek;

Data

Language:	Slovenian
Year of publishing:	2021
Typology:	2.11 - Undergraduate Thesis
Organization:	UL FMF - Faculty of Mathematics and Physics
Publisher:	[T. Mulej]
UDC:	512
COBISS:	78322435
Views:	959
Downloads:	87
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Normal matrices
Secondary abstract:	Matrix normality is one of the most interesting topics in linear algebra and matrix theory, since normal matrices have not only simple structures under unitary similarity but also many applications, that is why it has been done a great deal of work on them. There are $89$ different characteristic properties. In this thesis we chose $25$ of those characteristic properties and proved their equivalence to basic definition of normal matrices. We were also interested in how “close” are the matrices in terms of their eigenvalues. More interestingly, if a matrix is “perturbed” a little bit, how would the eigenvalues of the matrix change? In this thesis we present answers to these two questions if the matrices are normal.
Secondary keywords:	matrices;eigenvalues;eigenvectors;Schur decomposition;polar decomposition;Hoffman-Wielandt theorem;Sun theorem;
Type (COBISS):	Final seminar paper
Study programme:	0
Thesis comment:	Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Matematika - 1. stopnja
Pages:	27 str.
ID:	13505879