Kompleksne ortogonalne matrike

Hladnik, Maja (author), Slapar, Marko (mentor)

Kompleksne ortogonalne matrike so ortogonalne matrike, pri katerih dopustimo tudi kompleksne elemente. Ker ohranimo pogoj, ki velja za ortogonalne matrike, to niso unitarne matrike, prav tako lastnosti, ki veljajo za unitarne matrike, ne veljajo nujno za kompleksne ortogonalne matrike. V magistrskem delu bomo najprej povzeli osnovne pojme, ki jih bomo potrebovali v nadaljevanju, med drugim, kaj je diagonalizabilnost, Jordanova kletka in Jordanova matrika. Pokazali bomo, kdaj so matrike diagonalizabilne, ter predstavili nekatere lastnosti Jordanovih kletk. V nadaljevanju magistrskega dela bomo za normalne, simetrične in ortogonalne matrike pokazali, kdaj so diagonalizabilne in kdaj niso. V sklepnem delu magistrskega dela se bomo posvetili kompleksnim ortogonalnim matrikam ter pokazali nekaj lastnosti, ki veljajo zanje. Končni cilj magistrskega dela je pokazati, da je Jordanova kanonična forma kompleksnih matrik, ki so podobne kompleksnim ortogonalnim matrikam, direktna vsota Jordanovih kletk le treh različnih oblik.

• Language: Slovenian
• Keywords: diagonalizabilnost
• Work type: Master's thesis/paper
• Typology: 2.09 - Master's Thesis
• Organization: PEF - Faculty of Education
• Year: 2019
• PID: 20.500.12556/RUL-111425
• COBISS.SI-ID: 12611657
• Publication date in RUL: 02.10.2019

Secondary language

• Language: English
• Title: Complex orthogonal matrices

Abstract

Complex orthogonal matrices are orthogonal matrices with complex elements. Because the characterisation of complex orthogonal matrices remains the same as the characterisation of orthogonal matrices, these matrices are not unitary and do not have the same properties as unitary matrices. In this master's thesis we will first present some basic definitions which we will need later in the thesis, such as diagonalization, Jordan blocks and Jordan matrices. We will show when matrices are diagonalizable and some of the properties of Jordan blocks. We will continue by showing when normal, symmetric and orthogonal matrices are diagonalizable and when they are not. In the end we will look at complex orthogonal matrices and show some of the properties that they possess. The goal of this thesis is to show that the Jordan canonical form of a complex matrix, similar to complex orthogonal matrix, is a direct sum of Jordan blocks of only three different types.

• Keywords: diagonalization