Kanonična forma za kompleksne simetrične matrike

delo diplomskega seminarja

Anja Kozinc (Author), Klemen Šivic (Mentor)

Abstract

Realne simetrične matrike so diagonalizabilne, kar pa v splošnem za kompleksne simetrične matrike, ki so obravnavane v diplomski nalogi, ne velja. Kompleksna simetrična matrika je diagonalizabilna natanko tedaj, ko vsak lastni podprostor vsebuje ortonormirano bazo. Če obstaja lastni podprostor, katerega vsaka ortogonalna baza vsebuje kak izotropični vektor, matrike ne moremo diagonalizirati. V diplomski nalogi so izotropični vektorji podrobneje predstavljeni, saj vplivajo na diagonalizabilnost obravnavanih matrik. Poleg tega sta za matrike, ki niso diagonalizabilne, predstavljeni dve možni simetrični kanonični formi, katerima je vsaka kompleksna simetrična matrika ortogonalno podobna.

Keywords

matematika;kompleksne simetrične matrike;diagonalizabilnost;izotropični vektorji;kanonična forma;

Data

Language:	Slovenian
Year of publishing:	2019
Typology:	2.11 - Undergraduate Thesis
Organization:	UL FMF - Faculty of Mathematics and Physics
Publisher:	[A. Kozinc]
UDC:	512
COBISS:	18719833
Views:	1328
Downloads:	141
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Canonical form for complex symmetric matrices
Secondary abstract:	A real symmetric matrix can be diagonalised by an orthogonal transformation. This statement is not true, in general, for a symmetric matrix with complex elements, which is discussed in the present thesis. A complex symmetric matrix can be diagonalised by an orthogonal transformation, if and only if each eigenspace of the matrix has an orthonormal basis. If there is an eigenspace, where every orthogonal basis contains an isotropic vector, the matrix can not be diagonalised. In the present thesis isotropic vectors are presented in greater detail. Also, two possible symmetric canonical forms are obtained for the non-diagonalisable case.
Secondary keywords:	mathematics;complex symmetric matrices;diagonalisability;isotropic vector;canonical form;
Type (COBISS):	Final seminar paper
Study programme:	0
Embargo end date (OpenAIRE):	1970-01-01
Thesis comment:	Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Matematika - 1. stopnja
Pages:	36 str.
ID:	11218583