Normalne matrike : delo diplomskega seminarja

Mulej, Tim (author), Drnovšek, Roman (mentor)

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.

• Language: Slovenian
• Keywords: matrika, lastna vrednost, lastni vektor, Schurjev razcep, polarni razcep, Hoffman-Wielandtov izrek, Sunov izrek
• Work type: Bachelor thesis/paper
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2021
• PID: 20.500.12556/RUL-131038
• UDC: 512
• COBISS.SI-ID: 78322435
• Publication date in RUL: 22.09.2021

Secondary language

• Language: English
• Title: Normal matrices

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.

• Keywords: matrix, eigenvalues, eigenvectors, Schur decomposition, polar decomposition, Hoffman–Wielandt theorem, Sun theorem