Bilinearne i kvadratne forme

master's thesis

• Author: Kristina Halambek
• Mentor: Ozren Perše (mentor)
• Committee member: Ozren Perše (predsjednik povjerenstva)
• Committee member: Dražen Adamović (član povjerenstva)
• Committee member: Vjeran Hari (član povjerenstva)
• Committee member: Zvonimir Tutek (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2016-11-30, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Tri su glavna tipa bilinearnih formi: simetrične, unitarne i alternirajuće. U ovom diplomskom radu dajemo strukturne teoreme o matričnim prikazima ovih formi s obzirom na odgovarajuće baze. Rad između ostalog, prati standardni postupak rastava objekata na direktnu sumu prostih objekata, koliko je to moguće. Nakon prvog poglavlja, u kojem su definirani osnovni pojmovi potrebni za razumijevanje daljnjeg sadržaja rada, slijedi drugo poglavlje koje je podijeljeno u jedanaest potpoglavlja. Pobliže promatramo bilinearna te kvadratna preslikavanja, a također i bilinearne i kvadratne forme. Dokazujemo također i egzistenciju ortogonalne baze za simetrične bilinearne forme te donosimo dijagonalni prikaz te forme. Zatim promatramo simetrične bilinearne forme definirane nad uređenim poljem te iskazujemo i dokazujemo Sylvesterov teorem te opisujemo induktivni postupak, Gram-Schmidtov postupak ortogonalizacije. Donosimo i dokaz egzistencije ortogonalne baze za vektorski prostor na kojem je definirana hermitska forma, a nakon toga slijede spektralni teoremi, u hermitskom, unitarnom te simetričnom slučaju. Slijede alternirajuće bilinearne forme te uvođenje definicija hiperboličke ravnine te hiperboličkog prostora. Na kraju, iskazujemo te dokazujemo Wittov teorem te definiramo pojam Wittove grupe.

Abstract (english)

There are three major types of bilinear forms: symmetric, unitary, and alternating. In this diploma thesis, we present structural theorems giving normalized expressions for these forms with respect to suitable bases. We follow the standard pattern of decomposing an object into a direct sum of simple objects, insofar as possible. After the first chapter, in which the basic concepts necessary to understand the further contents of the work were defined, follows the second chapter which is divided into eleven subsections. We closely study bilinear and quadratic maps and also bilinear and quadratic forms. We also prove the existence of orthogonal basis for symmetric bilinear forms and present the diagonal representation of this form. Furthermore, we study bilinear symmetric forms defined on an ordered field, we state and prove Sylvester theorem and we describe the inductive process, Gram-Schmidt orthogonalization process. We present the proof of existence of orthogonal basis for a vector space in which the hermitian form is defined, followed by spectral theorems, in hermitian, unitary and symmetric cases. After that we study alternating bilinear forms and define hyperbolic plane and hyperbolic space. Finally, we state the Witt theorem and define the concept of Witt group.

• Keywords: bilinearne forme; kvadratne forme; hermitske forme; alternirajuće forme; Sylvesterov teorem; Gram-Schmidtov postupak ortogonalizacije; Wittov teorem; Wittova grupa
• Keywords (english): bilinear forms; quadratic forms; Hermitian forms; alternating forms; Sylvester theorem; Gram-Schmidt orthogonalization process; Witt theorem; Witt group
• Language: croatian
• URN:NBN: urn:nbn:hr:217:145897
• Study programme: Title: Mathematics and Computer Science Education; specializations in: Mathematics and Computer Science Education; Course: Mathematics and Computer Science Education; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra edukacije matematike i informatike (magistar/magistra edukacije matematike i informatike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2017-03-21 13:38:13