Posplošeni inverzi realnih matrik

magistrsko delo

Katja Mihelič (Author), Janko Marovt (Mentor)

Abstract

Inverz matrike je definiran za kvadratne nesingularne matrike. Velikokrat imamo opravka s pravokotnimi ali singularnimi matrikami, a vseeno potrebujemo matriko, ki se obnaša podobno kot inverz. Za take primere definiramo posplošeni inverz ali pseudoinverz. V uvodnem (prvem) poglavju magistrske naloge najprej predstavimo nekaj osnovnih pojmov in definicij, ki so potrebni za razumevanje nadaljnje vsebine. V osrednjem (drugem) poglavju definiramo Moore-Penroseov inverz, ki zadošča štirim pogojem, in si podrobno ogledamo njegove lastnosti. Raziščemo Moore-Penroseov inverz vsote in produkta matrik. Definiramo še posplošeni notranji inverz in inverz najmanjših kvadratov ter si pogledamo nekatere njune lastnosti. Zaključimo z računanjem vseh treh posplošenih inverzov. V zaključnem (tretjem) poglavju predstavimo uporabo posplošenih inverzov za reševanje sistemov linearnih enačb. Sisteme razdelimo na rešljive in nerešljive ter za nerešljive predstavimo metodo najmanjših kvadratov.

Keywords

posplošeni inverzi;pseudoinverzi;realne matrike;Moore-Penroseovi inverzi;posplošeni notranji inverzi;inverzi najmanjših kvadratov;sistemi linearnih enačb;metode najmanjših kvadratov;magistrska dela;

Data

Language:	Slovenian
Year of publishing:	2015
Typology:	2.09 - Master's Thesis
Organization:	UM FNM - Faculty of Natural Sciences and Mathematics
Publisher:	[K. Mihelič]
UDC:	512.643.43(043.2)
COBISS:	21547016
Views:	1163
Downloads:	118
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Generalized inverses of real matrices
Secondary abstract:	The inverse of a matrix is defined for all square nonsingular matrices. Sometimes we may have a rectangular matrix or a square singular matrix and still be in need of another matrix that in some ways behaves like the inverse. For such situations we define generalized inverse or pseudoinverse. The introductory chapter (first chapter) initially includes some basic terms and definitions that are needed for further understanding of the content. Second chapter is the central part of the master thesis. There we define Moore-Penrose inverse which satisfies four conditions, and take a close look at its properties. Furthermore we discover Moore-Penrose generalized inverse of the sum of matrices and the product of matrices. Additionally we define another two generalized inverses, the inner generalized inverse and the least squares inverse, and look at some of their properties. We conclude this chapter by calculating all three inverses. In the final chapter we present how to use generalized inverses for finding solutions to a system of linear equations. We divide systems into consistent and inconsistent. For inconsistent systems of linear equations we introduce the method of least squares solutions.
Secondary keywords:	generalised inverses;pseudoinverses;real matrices;Moore-Pernose inverses;inner generalized inverses;systems of linear equations;least squares solutions;master theses;
URN:	URN:SI:UM:
Type (COBISS):	Master's thesis/paper
Thesis comment:	Univ. v Mariboru, Fak. za naravoslovje in matematiko, Oddelek za matematiko in računalništvo
Pages:	91 f.
ID:	8756453