292 KB519 KB0 KB2005Cigler, GregorDrnovšek, RomanOmladič, Matjaž101 str., 30 cmCOBISSID:13609561URN:URN:NBN:SI:doc-SJ3H5QKJenG. Ciglervisokošolska delaArbeidsgiverforeningen Spektergroup theorymatematikamathematicsmatrične grupematrix theorypermutacijska matrikapermutation matrixpodobnostsimilarityspectrumspekterteorija grupteorija matriktriangularizationtrikotljivostGroups of matrices with prescribed spectrum| doctoral dissertation| doktorska disertacija| Grupe matrik s predpisanim spektrom|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 detailV 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 ?$n = 2,3,4,5$?TEXTvisokošolska delatheses and dissertationsSlovenian National E-content AggregatorNational and University Library of SloveniaUniverza v Ljubljani, Fakulteta za matematiko in fiziko