{"?xml":{"@version":"1.0"},"edm:RDF":{"@xmlns:dc":"http://purl.org/dc/elements/1.1/","@xmlns:edm":"http://www.europeana.eu/schemas/edm/","@xmlns:wgs84_pos":"http://www.w3.org/2003/01/geo/wgs84_pos","@xmlns:foaf":"http://xmlns.com/foaf/0.1/","@xmlns:rdaGr2":"http://rdvocab.info/ElementsGr2","@xmlns:oai":"http://www.openarchives.org/OAI/2.0/","@xmlns:owl":"http://www.w3.org/2002/07/owl#","@xmlns:rdf":"http://www.w3.org/1999/02/22-rdf-syntax-ns#","@xmlns:ore":"http://www.openarchives.org/ore/terms/","@xmlns:skos":"http://www.w3.org/2004/02/skos/core#","@xmlns:dcterms":"http://purl.org/dc/terms/","edm:WebResource":[{"@rdf:about":"http://www.dlib.si/stream/URN:NBN:SI:DOC-SJ3H5QKJ/b52144a0-4601-49df-9ce8-fe98dbfcc6fa/HTML","dcterms:extent":"292 KB"},{"@rdf:about":"http://www.dlib.si/stream/URN:NBN:SI:DOC-SJ3H5QKJ/6f7854e7-7b8d-4635-a0bc-67c7acd5fbda/PDF","dcterms:extent":"519 KB"},{"@rdf:about":"http://www.dlib.si/stream/URN:NBN:SI:DOC-SJ3H5QKJ/7f1b8826-56da-442b-9060-e354c601383e/TEXT","dcterms:extent":"0 KB"}],"edm:ProvidedCHO":{"@rdf:about":"URN:NBN:SI:DOC-SJ3H5QKJ","dcterms:issued":"2005","dc:creator":"Cigler, Gregor","dc:contributor":["Drnovšek, Roman","Omladič, Matjaž"],"dc:format":{"@xml:lang":"sl","#text":"101 str., 30 cm"},"dc:identifier":["COBISSID:13609561","URN:URN:NBN:SI:doc-SJ3H5QKJ"],"dc:language":"en","dc:publisher":{"@xml:lang":"sl","#text":"G. Cigler"},"dc:source":{"@xml:lang":"sl","#text":"visokošolska dela"},"dc:subject":[{"@xml:lang":"en","#text":"Arbeidsgiverforeningen Spekter"},{"@xml:lang":"en","#text":"group theory"},{"@xml:lang":"sl","#text":"matematika"},{"@xml:lang":"en","#text":"mathematics"},{"@xml:lang":"sl","#text":"matrične grupe"},{"@xml:lang":"en","#text":"matrix theory"},{"@xml:lang":"sl","#text":"permutacijska matrika"},{"@xml:lang":"en","#text":"permutation matrix"},{"@xml:lang":"sl","#text":"podobnost"},{"@xml:lang":"en","#text":"similarity"},{"@xml:lang":"en","#text":"spectrum"},{"@xml:lang":"sl","#text":"spekter"},{"@xml:lang":"sl","#text":"teorija grup"},{"@xml:lang":"sl","#text":"teorija matrik"},{"@xml:lang":"en","#text":"triangularization"},{"@xml:lang":"sl","#text":"trikotljivost"},{"@rdf:resource":"http://www.wikidata.org/entity/Q874429"}],"dc:title":{"@xml:lang":"sl","#text":"Groups of matrices with prescribed spectrum| doctoral dissertation| doktorska disertacija| Grupe matrik s predpisanim spektrom|"},"dc:description":[{"@xml:lang":"sl","#text":"The general form of the problem that we discuss in this work is the following. Let ?$\\mathcal{G}$? be a (semi)group of ?$n\\times n$? matrices over the field ?$\\mathbb{F}$? such that each matrix from ?$\\mathcal{G}$? is (individually) similar to a matrix with a given property ?$\\mathcal{P}$?. Is then the (semi)group ?$\\mathcal{G}$? simultaneously similar to a (semi)group of matrices all having the property ?$\\mathcal{P}$?, i.e., can we find an invertible matrix ?$S \\in {\\mathbb{F}}^{n\\times n}$? such that for all ?$X \\in \\mathcal{G}$? the matrix ?$SXS^{-1}$? has the property ?$\\mathcal{P}$?? When the answer is negative in general, we search for additional assumptions under which the (semi)group ?$\\mathcal{G}$? is simultaneously similar to a desired (semi)group. In Chapters 2 and 3 we consider triangularizability of matrix (semi)groups. In this case a matrix has the property ?$\\mathcal{P}$?, if it is upper triangular and its spectrum satisfy some additional conditions. If ?$\\mathcal{G}$? is a matrix semigroup which is triangularized, diagonal entries on a fixed position form a subsemigroup of the multiplicative group ?${\\mathbb{F}} \\setminus \\{0\\}$?. In Chapter 2 we study the triangularizability under the assumption that the union of the spectra of all matrices from ?$\\mathcal{G}$? forms a group ?$\\Gamma$?. When ?$\\Gamma = \\{1\\}$? is the trivial group, the well-known Kolchin''s theorem gives the affirmative answer to our problem: Every semigroup of unipotent matrices is triangularizable. We investigate the case where ?$\\Gamma$? is a finite group and we show that Kolchin''s theorem extends only to the case where ?$\\Gamma = \\{1,-1\\}$?. We give counterexamples for groups ?$\\Gamma$? not contained in the group ?$\\{1,-1\\}$?. In the search for further extensions of Kolchin''s theorem in Chapter 3 we introduce ?$p$?-property, which is some kind of independency condition of the eigenvalues. We investigate more closely groups of monomial matrices with this property. The main theorem of this chapter is a generalization of Kolchin''s theorem to the groups with ?$2$?-property. Chapter 4 is dedicated to the permutation-like groups, i.e., the finite groups ?$\\mathcal{G} \\subset \\CC^{n \\times n}$? such that any matrix ?$X \\in \\mathcal{G}$? is similar to a permutation matrix. In this case a matrix has the property ?$\\mathcal{P}$?, if it is a permutation matrix. Since a matrix similar to a permutation matrix is determined by its spectrum, we could describe this property in terms of the spectrum, but the previous description is more transparent. We deal with the question when a permutation-like group is simultaneously similar to a group of permutation matrices. Various examples in this chapter show that in general a permutation-like group does not have to be simultaneously similar to a group of permutation matrices. In fact, there are counterexamples for every ?$n \\ge 6$?. The low-dimensional cases ?$n = 2,3,4,5$? are investigated in detail"},{"@xml:lang":"sl","#text":"V delu obravnavamo naslednji splošni problem. Naj bo ?$\\mathcal{G}$? taka (pol)grupa ?$n\\times n$? matrik nad poljem ?$\\mathbb{F}$?, da je vsaka matrika iz ?$\\mathcal{G}$? podobna kaki matriki z dano lastnostjo ?$\\mathcal{P}$?. Ali je tedaj (pol)grupa ?$\\mathcal{G}$? v celoti podobna kaki (pol)grupi matrik, ki imajo vse lastnost ?$\\mathcal{P}$?, tj., ali obstaja taka obrnljiva matrika ?$S \\in {\\mathbb{F}}^{n\\times n}$?, da ima za vsako matriko ?$X \\in \\mathcal{G}$? matrika ?$SXS^{-1}$? lastnost ?$\\mathcal{P}$?? Če je odgovor na to splošno vprašanje negativen, iščemo dodatne zadostne pogoje za to, da je (pol)grupa ?$\\mathcal{G}$? simultano podobna željeni (pol)grupi. V drugem in tretjem poglavju se ukvarjamo s problemom trikotljivosti matričnih (pol)grup. V tem primeru ima matrika lastnost ?$\\mathcal{P}$?, če je zgornjetrikotna in njen spekter zadošča nekim dodatnim pogojem. Če je ?$\\mathcal{G}$? polgrupa iz trikotnih matrik, diagonalni elementi na izbranem mestu tvorijo podpolgrupo multiplikativne grupe ?${\\mathbb{F}} \\setminus \\{0\\}$?. V drugem poglavju se ukvarjamo s trikotljivostjo pri predpostavki, da unija spektrov vseh matrik iz ?$\\mathcal{G}$? tvori neko grupo ?$\\Gamma$?. Če je ?$\\Gamma = \\{1\\}$? trivialna grupa, nam znani Kolčinov izrek da pritrdilen odgovor: vsaka polgrupa unipotentnih matrik je trikotljiva. V našem sem primeru privzamemo, da je ?$\\Gamma$? končna grupa in dokažemo, da je Kolčinov izrek možno raširiti le na grupo ?$\\Gamma = \\{1,-1\\}$?. Za vse grupe ?$\\Gamma$?, ki niso vsebovane v ?$\\{1,-1\\}$? konstruiramo protiprimere. V tretjem poglavju vpeljemo ?$p$?-lastnost, ki je neke vrste neodvisnost lastnih vrednosti dane matrike. Natančneje preučimo monomialne grupe s to lastnostjo in Kolčinov izrek razširimo na grupe z -lastnostjo. Četrto poglavje je posvečeno lokalno permutacijskim grupam matrik, tj., takim končnim grupam ?$\\mathcal{G} \\subset \\CC^{n \\times n}$?, da je vsaka matrika ?$X \\in \\mathcal{G}$? podobna neki permutacijski matriki. V tem primeru ima matrika lastnost ?$\\mathcal{P}$?, če je permutacijska matrika. Ker je vsaka permutacijska matrika določena s svojim spektrom, lahko to lastnost opišemo tudi s pomočjo spektra, venderje prvotni opis vsekakor bolj nazoren. Ukvarjamo se z vprašanjem, kdaj je neka lokalno permutacijska grupa simultano podobna kaki grupi iz permutacijskih matrik. Z raznimi primeri pokažemo, da to v splošnem ni možno. Protiprimeri obstajajo v vseh dimenzijah ?$n \\ge 6$?. Natančno raziščemo dimenzije ?$n = 2,3,4,5$?"}],"edm:type":"TEXT","dc:type":[{"@xml:lang":"sl","#text":"visokošolska dela"},{"@xml:lang":"en","#text":"theses and dissertations"},{"@rdf:resource":"http://www.wikidata.org/entity/Q1266946"}]},"ore:Aggregation":{"@rdf:about":"http://www.dlib.si/?URN=URN:NBN:SI:DOC-SJ3H5QKJ","edm:aggregatedCHO":{"@rdf:resource":"URN:NBN:SI:DOC-SJ3H5QKJ"},"edm:isShownBy":{"@rdf:resource":"http://www.dlib.si/stream/URN:NBN:SI:DOC-SJ3H5QKJ/6f7854e7-7b8d-4635-a0bc-67c7acd5fbda/PDF"},"edm:rights":{"@rdf:resource":"http://rightsstatements.org/vocab/InC/1.0/"},"edm:provider":"Slovenian National E-content Aggregator","edm:intermediateProvider":{"@xml:lang":"en","#text":"National and University Library of Slovenia"},"edm:dataProvider":{"@xml:lang":"sl","#text":"Univerza v Ljubljani, Fakulteta za matematiko in fiziko"},"edm:object":{"@rdf:resource":"http://www.dlib.si/streamdb/URN:NBN:SI:DOC-SJ3H5QKJ/maxi/edm"},"edm:isShownAt":{"@rdf:resource":"http://www.dlib.si/details/URN:NBN:SI:DOC-SJ3H5QKJ"}}}}