Invariantni podprostori linearnih operatorjev nad R

Polc, Katarina (author), Malnič, Aleksander (mentor)

V nalogi ločimo operatorje, ki s svojim delovanjem na vektorski prostor porodijo razpad prostora na premo vsoto samih, za izbrani operator minimalnih invariantnih podprostorov. Takšne operatorje imenujemo povsem reducibilni. V delu definiramo invariantne podprostore. Inducirane operatorje, ki delujejo nad njimi, preučimo vsaj do te mere, da lahko vpeljemo lastne ter korenske podprostore, za tem pa presodimo, v kakšnih primerih sta ta dva enaka in kakšne so posledice tega. S tem ločimo operatorje, ki porodijo razpad prostora na same enorazsežne, za dani operator invariantne podprostore, ter operatorje, katerih razpad prostora, na katere delujejo, ni tak. Ob tem vpeljemo vse potrebno orodje za opis takega razcepa.

• Language: Slovenian
• Keywords: linearni operator
• Work type: Bachelor thesis/paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: PEF - Faculty of Education
• Year: 2017
• PID: 20.500.12556/RUL-95576
• COBISS.SI-ID: 11718473
• Publication date in RUL: 21.09.2017

Secondary language

• Language: English
• Title: Invariant subspaces of linear operators over R

Abstract

In the thesis, we separate operators which have an effect on a vector space in such a way that they cause a decomposition of that space into a direct sum of minimal invariant subspaces. These kinds of operators are completely reducible. Invariant subspaces are then defined. We define induced operators that affect them to such extent that we can introduce their eigenspaces and root subspaces (generalized eigenspaces) and judge in which cases these are the same and what the consequences of that fact are. By using this procedure, we can separate the operators on those which cause a decomposition of space on only one-dimensional invariant subspaces and on those of which the decomposition of the space they affect on is different. The end result of this thesis is a treatment of the tool used to describe that kind of decomposition.

• Keywords: linear operator