Kanonična forma za kompleksne simetrične matrike : delo diplomskega seminarja

Kozinc, Anja (author), Šivic, Klemen (mentor)

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.

• Language: Slovenian
• Keywords: kompleksna simetrična matrika, diagonalizabilnost, izotropični vektor, kanonična forma
• Work type: Final seminar paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2019
• PID: 20.500.12556/RUL-109772
• UDC: 512
• COBISS.SI-ID: 18719833
• Publication date in RUL: 08.09.2019

Secondary language

• Language: English
• Title: Canonical form for complex symmetric matrices

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.

• Keywords: complex symmetric matrix, diagonalisability, isotropic vector, canonical form