Vizualizacija polarnega razcepa linearnih transformacij : diplomsko delo

Hrvatin, Erik (author), Kuzman, Boštjan (mentor)

V diplomskem delu obravnavamo polarni razcep za realne kvadratne matrike. Gre za produkt pozitivno definitne in ortogonalne matrike, s katerimi se tudi nekoliko podrobneje srečamo. V delu dokažemo, da polarni razcep vedno obstaja in je enolično določen za obrnljive matrike. Posebej izpeljemo ustrezne formule za ortogonalne in pozitivno definitne matrike dimenzije 2x2, prav tako pa tudi eksplicitno formulo za polarni razcep matrik dimenzije 2x2 s pomočjo katere lahko ugotovimo, kdaj ima matrika polarni razcep nad poljem racionalnih števil. Matrike si lahko predstavljamo tudi kot linearne transformacije ravnine, kar ilustriramo s slikami in interaktivnimi apleti, ki smo jih izdelali v programu GeoGebra.

• Language: Slovenian
• Keywords: matrika, pozitivno definitna matrika, ortogonalna matrika, polarni razcep, linearna transformacija, vizualizacija
• Work type: Bachelor thesis/paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: PEF - Faculty of Education
• Publisher: [E. Hrvatin]
• Year: 2017
• Number of pages: 42 str.
• PID: 20.500.12556/RUL-95153
• UDC: 512.643.12(043.2)
• COBISS.SI-ID: 11695433
• Publication date in RUL: 19.09.2017

Secondary language

• Language: English
• Title: Visualisation of polar decomposition of linear transformations

Abstract

The thesis aims at addressing the polar decomposition of a real square matrix. This is the product of a positive-definite matrix and an orthogonal matrix that are discussed in more detail as well. It is shown and proved in the thesis that the polar decomposition always exists and it is unique for invertible matrices. The adequate formula for orthogonal and positive-definite 2x2 matrices and the explicit formula for the polar decomposition of 2x2 matrices are derived. The latter can be used to help us determine when a polar decomposition of a matrix has rational coefficients. Matrices can also be seen as linear transformations of the plane. They can be visually represented with images or interactive applets that were developed using the GeoGebra program.

• Keywords: mathematics, matematika