magistrsko delo
Gregor Podlogar (Author), Igor Klep (Mentor), Matej Brešar (Co-mentor)

Abstract

V delu obravnavamo invariante $m$-teric $n \times n$ matrik $X_1, \ldots, X_m$ glede na hkratno konjugacijo. Pokažemo, da je vsako invarianto možno zapisati z matričnimi sledmi. Obravnavamo tudi konkomitante in pokažemo, da so kot algebra nad invariantami generirane s projekcijami na $X_i$. Vpeljemo polinome s sledmi in centralne polinome s sledmi. Prvi služijo zapisu konkomitant, drugi pa zapisu invariant. Spoznamo tudi identitete s sledmi in centralne identitete s sledmi, tj. polinome, ki določajo ničelno konkomitanto oziroma invarianto. Pokažemo, da je vsaka identiteta posledica Cayley-Hamiltonovega izreka.

Keywords

polinomske invariante matrik;konkomitante;polinomi s sledmi;identitete s sledmi;

Data

Language: Slovenian
Year of publishing:
Typology: 2.09 - Master's Thesis
Organization: UL FMF - Faculty of Mathematics and Physics
Publisher: [G. Podlogar]
UDC: 512.5
COBISS: 18431833 Link will open in a new window
Views: 1178
Downloads: 460
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Other data

Secondary language: English
Secondary title: Matrix invariants and trace identities
Secondary abstract: We consider invariants of $m$-tuples of $n \times n$ matrices $X_1, \ldots, X_m$ under simultaneous conjugation. We show that any invariant can be expressed using the trace. We also consider concomitants and describe them as an algebra over the invariants generated by the projections on $X_i$. For the purpose of describing invariants and concomitants we introduce trace polynomials. We consider trace identities, i.e. trace polynomials describing the zero invariant or concomitant. We show that any identity is a consequence of the Cayley-Hamilton theorem.
Secondary keywords: polynomial matrix invariants;concomitants;trace polynomials;trace identities;
Type (COBISS): Master's thesis/paper
Study programme: 0
Thesis comment: Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Matematika - 2. stopnja
Pages: IX, 41 str.
ID: 10958870
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