Popolnoma pozitiven rang simetričnih matrik : delo diplomskega seminarja

Kravanja, Ana Marija (avtor), Oblak, Polona (mentor)

V delu diplomskega seminarja bomo obravnavali popolnoma pozitivne matrike in njihov popolnoma pozitiven rang. Dva s tem povezana problema o določanju popolne pozitivnosti matrike in izračunljivosti njenega popolnoma pozitivnega ranga sta še vedno odprta, zato bomo sprva predstavili glavne definicije in rezultate na temo popolnoma pozitivnih matrik. Ogledali si bomo povezavo med popolnoma pozitivnimi matrikami, M-matrikami in diagonalno dominantnimi matrikami ter geometrijski pogled na popolnoma pozitiven rang. Nato si bomo pogledali alternativen postopek iskanja omejitev popolnoma pozitivnega ranga matrik s pomočjo teorije grafov. Natančneje, podali bomo lastnosti pripadajočih grafov, ki omejijo popolnoma pozitiven rang matrik pripadajočih vzorcev.

• Jezik: Slovenski jezik
• Ključne besede: popolnoma pozitivne matrike, popolnoma pozitiven rang, pozitivno semidefinitne matrike, M-matrike, konveksni stožci, pokritje grafa s klikami
• Vrsta gradiva: Diplomsko delo/naloga
• Organizacija: FMF - Fakulteta za matematiko in fiziko
• Leto izida: 2020
• PID: 20.500.12556/RUL-120459
• UDK: 512
• COBISS.SI-ID: 58094851
• Datum objave v RUL: 20.09.2020

Sekundarni jezik

• Jezik: Angleški jezik
• Naslov: The completely positive rank of symmetric matrices

Izvleček

In this thesis, we will investigate the completely positive matrices and their completely positive rank. The problems of determining whether a matrix is completely positive and computing its completely positive rank are still open. We will first present the main definitions and known results of this topic. We will also present M-matrices, diagonally dominant matrices, and discuss the geometric approach to complete positivity. Furthermore, we will take a look at the alternative procedure of finding constraints of the completely positive rank of matrices using graph theory. In particular, we will define the characteristics of the corresponding graphs which bound the completely positive rank of the matrix.

• Ključne besede: completely positive matrices, completely positive rank, positive semidefinite matrices, M-matrices, convex cones, clique covering number