Racionalna kanonična forma matrike

Šimon, Kaja

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Title:	Racionalna kanonična forma matrike
Authors:	ID Šimon, Kaja (Author) ID Benkovič, Dominik (Mentor) More about this mentor...
Files:	MAG_Simon_Kaja_2025.pdf (1,26 MB) MD5: 35E343D38AC1B7A3891E97B27951B55E
Language:	Slovenian
Work type:	Master's thesis/paper
Typology:	2.09 - Master's Thesis
Organization:	FNM - Faculty of Natural Sciences and Mathematics
Abstract:	V magistrskem delu obravnavamo, kako lahko linearni operator T: V -> V, kjer je V končno razsežen vektorski prostor nad poljem F, v ustrezni bazi predstavimo s posebno matriko z elementi iz polja F, imenovano racionalna kanonična forma. Ta zapis temelji na posebnem rezultatu, ki pravi, da lahko prostor V zapišemo kot direktno vsoto T-cikličnih podprostorov. V prvem delu predstavimo izsledke o polinomih, ki jih bomo potrebovali v nadaljevanju. Sledi pregled temeljnih pojmov, kot so lastne vrednosti, karakteristični in minimalni polinom, invariantni podprostori, direktne vsote ter izrek o primarni razčlenitvi. V nadaljevanju obravnavamo izrek o ciklični dekompoziciji, ki omogoča, da linearni operator predstavimo z matriko v racionalni kanonični obliki. Ta je sestavljena kot direktna vsota pridruženih matrik moničnih polinomov, imenovanih invariantni faktorji. Dotaknemo se tudi Jordanove oblike, ki opisuje matrike linearnih operatorjev z značilnostjo, da se njihov karakteristični polinom popolnoma razcepi nad obravnavanim poljem. V zaključnem delu predstavimo še postopek za izračun racionalne kanonične forme s pomočjo elementarnih podobnostnih transformacij.
Keywords:	matrika, racionalna kanonična forma, ciklična dekompozicija, pridružena matrika, invariantni faktorji, Jordanova forma
Place of publishing:	Maribor
Publisher:	[K. Šimon]
Year of publishing:	2025
PID:	20.500.12556/DKUM-94642
UDC:	512.643(043.2)
COBISS.SI-ID:	253419011
Publication date in DKUM:	16.10.2025
Views:	164
Downloads:	64
Metadata:
Categories:	FNM
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Licences

License:	CC BY-NC-ND 4.0, Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International

Link:	http://creativecommons.org/licenses/by-nc-nd/4.0/
Description:	The most restrictive Creative Commons license. This only allows people to download and share the work for no commercial gain and for no other purposes.
Licensing start date:	23.08.2025

Secondary language

Language:	English
Title:	The rational canonical form of a matrix
Abstract:	In this master's thesis, we discuss how a linear operator T: V -> V, where V is a finite-dimensional vector space over a field F, can be represented in a suitable basis by a special matrix with entries from the field F, called the rational canonical form. This representation is based on a particular result which states that the space V can be written as a direct sum of T-cyclic subspaces. In the first part, we present results on polynomials that we will need later on. This is followed by a review of fundamental concepts such as eigenvalues, the characteristic and minimal polynomial, invariant subspaces, direct sums, and the primary decomposition theorem. Next, we discuss the cyclic decomposition theorem, which allows us to represent a linear operator by a matrix in rational canonical form. This form is constructed as a direct sum of companion matrices of monic polynomials, called invariant factors. We also touch upon the Jordan form, which describes matrices of linear operators whose characteristic polynomial splits completely over the field in question. In the final part, we present the procedure for computing the rational canonical form using elementary similarity transformations.
Keywords:	matrix, rational canonical form, cyclic decomposition, companion matrix, invariant factors, Jordan form

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