Holomorfne matrične funkcije

delo diplomskega seminarja

Jaka Vrhovnik (Author), Miran Černe (Mentor)

Abstract

Za kvadratno matriko $A$ in funkcijo $f$, holomorfno na okolici spektra matrike $A$, lahko definiramo matriko $f(A)$. V delu predstavimo definicijo preko resolvente, tj. inverza matrike $zI - A$, ki je matrika holomorfnih funkcij na komplementu spektra matrike $A$, in pokažemo, da se tako definirane matrične funkcije lepo obnašajo za seštevanje, množenje in komponiranje funkcij. Nato poiščemo potrebne in zadostne pogoje, da dana holomorfna funkcija inducira surjektivno preslikavo na ustrezno podmnožico kvadratnih matrik, in raziščemo, pod katerimi pogoji ima matrika polinomov ali celih funkcij dobro definiran logaritem.

Keywords

primarna matrična funkcija;resolventa;Cauchyjeva formula;surjektivnost matričnih funkcij;logaritem matrike;

Data

Language:	Slovenian
Year of publishing:	2024
Typology:	2.11 - Undergraduate Thesis
Organization:	UL FMF - Faculty of Mathematics and Physics
Publisher:	[J. Vrhovnik]
UDC:	517.9
COBISS:	207595267
Views:	68
Downloads:	12
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Holomorphic Matrix Functions
Secondary abstract:	For a square matrix $A$ and a function $f$, holomorphic on a neighbourhood of the spectrum of the matrix $A$, we can define the matrix $f(A)$. In this thesis, we present the definition via the resolvent, i. e. the inverse of the matrix $zI - A$, which is a matrix of holomorphic functions on the complement of the spectrum of the matrix $A$, and we show that matrix functions defined in this way behave well under addition, multiplication and composition of functions. We then seek necessary and sufficient conditions for a given holomorphic function to induce a surjective mapping on the appropriate subset of square matrices, and we explore under what conditions a matrix of polynomials or entire functions has a well-defined logarithm.
Secondary keywords:	primary matrix function;resolvent;Cauchy integral formula;surjectivity of matrix functions;logarithm of a matrix;
Type (COBISS):	Final seminar paper
Study programme:	0
Embargo end date (OpenAIRE):	1970-01-01
Thesis comment:	Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Matematika - 1. stopnja
Pages:	29 str.
ID:	24939872