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Title:
Posplošeni inverzi realnih matrik
Authors:
ID
Mihelič, Katja
(Author)
ID
Marovt, Janko
(Mentor)
More about this mentor...
Files:
MAG_Mihelic_Katja_2015.pdf
(679,21 KB)
MD5: 51FEC0623CC2DA9C7A1A30098BE30A84
Language:
Slovenian
Work type:
Master's thesis/paper
Typology:
2.09 - Master's Thesis
Organization:
FNM - Faculty of Natural Sciences and Mathematics
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 inverz
,
pseudoinverz
,
realne matrike
,
Moore-Penroseov inverz
,
posplošeni notranji inverz
,
inverz najmanjših kvadratov
,
sistemi linearnih enačb
,
metoda najmanjših kvadratov
Place of publishing:
Maribor
Publisher:
[K. Mihelič]
Year of publishing:
2015
PID:
20.500.12556/DKUM-48241
UDC:
512.643.43(043.2)
COBISS.SI-ID:
21547016
NUK URN:
URN:SI:UM:DK:TOXBCHKY
Publication date in DKUM:
07.10.2015
Views:
1758
Downloads:
164
Metadata:
Categories:
FNM
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:
MIHELIČ, Katja, 2015,
Posplošeni inverzi realnih matrik
[online]. Master’s thesis. Maribor : K. Mihelič. [Accessed 4 April 2025]. Retrieved from: https://dk.um.si/IzpisGradiva.php?lang=eng&id=48241
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Secondary language
Language:
English
Title:
Generalized inverses of real matrices
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.
Keywords:
generalized inverse
,
pseudoinverse
,
real matrices
,
Moore-Penrose inverse
,
inner generalized inverse
,
least square inverse
,
systems of linear equations
,
least squares solutions
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