Invariante matrik in identitete s sledmi : magistrsko delo

Podlogar, Gregor (author), Klep, Igor (mentor), Brešar, Matej (comentor)

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.

• Language: Slovenian
• Keywords: polinomske invariante matrik, konkomitante, polinomi s sledmi, identitete s sledmi
• Work type: Master's thesis/paper
• Typology: 2.09 - Master's Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2018
• PID: 20.500.12556/RUL-102841
• UDC: 512
• COBISS.SI-ID: 18431833
• Publication date in RUL: 09.09.2018

Secondary language

• Language: English
• Title: Matrix invariants and trace identities

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.

• Keywords: polynomial matrix invariants, concomitants, trace polynomials, trace identities