20.500.12556/DKUM-48241 Posplošeni inverzi realnih matrik Generalized inverses of real matrices 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. 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. posplošeni inverz pseudoinverz realne matrike Moore-Penroseov inverz posplošeni notranji inverz inverz najmanjših kvadratov sistemi linearnih enačb metoda najmanjših kvadratov generalized inverse pseudoinverse real matrices Moore-Penrose inverse inner generalized inverse least square inverse systems of linear equations least squares solutions true true false [K. Mihelič] Slovenski jezik Angleški jezik Magistrsko delo/naloga 2015-06-02 20:12:17 2015-10-07 19:12:34 2022-05-24 03:06:46 2021-05-08 03:36:33 2015 0 Maribor 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 512.643.43(043.2) 21547016 URN:SI:UM:DK:TOXBCHKY MAG_Mihelic_Katja_2015.pdf MAG_Mihelic_Katja_2015.pdf 1 51FEC0623CC2DA9C7A1A30098BE30A84 dcd2f783fe97e2ff0ec20d09d3db620a02a55785992c4fe4bf8a3fdb201bf4f6 ab47105f-7c0b-11eb-bb7a-00155d0001ca https://dk.um.si/Dokument.php?lang=slv&id=72193 Fakulteta za naravoslovje in matematiko 0 0 0