Razcepi matrik in njihova uporaba

magistrsko delo

Jan Kern (Author), Luka Boc Thaler (Mentor)

Abstract

Razcep matrike nam omogoči, da iz kompleksne matrike dobimo produkt dveh ali več enostavnejših matrik. To storimo z namenom, da lahko nekatere matrične operacije lažje izvedemo na razstavljenih matrikah kot na prvotni matriki. Obstaja mnogo različnih vrst razcepov matrik, ki imajo različne lastnosti in implikacije v praksi. V tem magistrskem delu bo predstavljenih pet najbolj splošnih in osnovnih razcepov matrik in njihove lastnosti. V delu bomo pokazali, kako z uporabo različnih razcepov matrik lahko rešujemo sisteme linearnih enačb in primerjali njihovo učinkovitost s standardno Gaussovo eliminacijsko metodo. Nadalje bomo predstavili tudi linearni problem najmanjših kvadratov, kjer se bomo s pomočjo razcepov matrik lotili reševanja predoločenega sistema linearnih enačb. V obeh primerih bomo med seboj primerjali učinkovitost uporabe posameznega razcepa glede na dan sistem enačb. V zadnjem delu pa bo na preprost način predstavljena uporaba posameznega razcepa na različnih znanstvenih področjih, s čimer bomo prikazali širino aplikativnosti razcepov matrik.

Keywords

razcep matrike;diagonalizacija;singularni razcep;LU razcep;razcep Choleskega;QR razcep;

Data

Language:	Slovenian
Year of publishing:	2023
Typology:	2.09 - Master's Thesis
Organization:	UL PEF - Faculty of Education
Publisher:	[J. Kern]
UDC:	51(043.2)
COBISS:	168495875
Views:	12
Downloads:	3
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Matrix decomposition and its applications
Secondary abstract:	Matrix factorization allows us to break down a complex matrix into a product of two or more simpler matrices. This is done with the intent of simplifying certain matrix operations, which can be more easily carried out on the factored matrices than on the original matrix. There are many different types of matrix factorizations, each with distinct properties and implications in practice. In this master's thesis, we will introduce five of the most common and fundamental matrix factorizations and their properties. We will demonstrate how using various matrix factorizations can solve systems of linear equations and compare their efficiency with the standard Gaussian elimination method. Additionally, we will introduce the linear least squares problem, where we will show, how to find the solution of overdetermined systems of linear equations using matrix factorizations. In both cases, we will compare the efficiency of using each type of factorization depending on the given system of equations. In the final part, we will present in a straightforward manner the application of each type of factorization across various scientific fields and with that, demonstrating the broad applicability of matrix factorizations.
Secondary keywords:	matrix decomposition;diagonalization;singular value decomposition;LU decomposition;Cholesky decomposition;QR decomposition;Matrike (matematika);Univerzitetna in visokošolska dela;
Type (COBISS):	Master's thesis/paper
Study programme:	0
Embargo end date (OpenAIRE):	1970-01-01
Thesis comment:	Univ. v Ljubljani, Pedagoška fak., Poučevanje
Pages:	1 spletni vir (1 datoteka PDF (74 str.))
DOI:	20.500.12556/RUL-151607
ID:	20202181