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V magistrskem delu je obravnavan problem določitve cp-ranga dane popolnoma pozitivne matrike. Uvodoma so opisane osnovne lastnosti pozitivno semidefinitnih matrik in predstavljeni so konveksni stožci evklidskega prostora V. V osrednjem delu se osredotočimo na popolnoma pozitivne matrike. Matrika A je popolnoma pozitivna, če jo lahko zapišemo kot A=BB^{T} za neko nenegativno matriko B. Dokažemo osnovne lastnosti popolnoma pozitivnih matrik ter definiramo diagonalno dominantne in primerjalne matrike. Delo zaključimo z obravnavo problema določitve cp-ranga popolnoma pozitivne matrike. Obravnavamo primer za matrike manjše velikosti ter določimo zgornjo mejo za cp-rang matrike danega ranga in matrike dane velikosti. In the master thesis the problem of determining the cp-rank of a given completely positive matrix is discussed. In the introduction the basic properties of positive semidefinite matrices are described and convex cones in euclidean space V are presented. In the main part we focus on completely positive matrices. Matrix A is completely positive if it can be decomposed as A=BB^{T}, where B is a nonnegative matrix. We prove the basic properties of totally positive matrices and define diagonally dominant and comparative matrix. The thesis is concluded with a discussion of a problem of determining the cp-rank of a completely positive matrix. We consider a case of a matrix of a smaller size and set an upper bound for cp-rank matrix of a given rank and a matrix of a given order.

Glavna tema magistrske naloge so popolnoma pozitivne matrike, ki so posebni primer pozitivno semidefinitnih matrik. Vsaka realna pozitivno semidefinitna matrika A se lahko zapiše kot A=BB^T, kjer je B realna matrika. V primeru, da je B nenegativna matrika, je matrika A popolnoma pozitivna. Na začetku predstavimo osnovne pojme in definicije realnih matrik, s poudarkom na pozitivno semidefinitnih matrikah. Podamo nekaj primerov in dokažemo osnovne lastnosti teh matrik. V nadaljevanju obravnavamo popolnoma pozitivne matrike. Definiramo Hadamardov in Kroneckerjev produkt ter dokažemo, da sta oba produkta popolnoma pozitivnih matrik popolnoma pozitivni matriki. Spoznamo eno izmed metod, s katero pokažemo, da je dvojno nenegativna matrika popolnoma pozitivna. Definiramo pojem konveksni stožec ter dokažemo, da je množica popolnoma pozitivnih matrik zaprt konveksni stožec. Na algebraični in geometrijski način dokažemo, da so t.i. majhne matrike popolnoma pozitivna. Nazadnje obravnavamo diagonalno dominantne matrike ter dokažemo, da so nenegativne simetrične diagonalno dominantne matrike popolnoma pozitivne. Prav tako definiramo primerjalno matriko in dokažemo, da je matrika A popolnoma pozitivna, če je simetrična nenegativna matrika ter je njena primerjalna matrika pozitivno semidefinitna. The main topic of the master thesis are completely positive matrices, which are the special case of a positive semidefinite matrix. Every real positive semidefinite matrix A can be written in the form A=BB^T, where B is a real matrix. In the case of a nonnegative matrix B the matrix A is completely positive. The first chapter includes some basic terms and definitions of specific real matrices with an emphasis on positive semidefinite matrices. We present some examples and prove the basic properties of these matrices. In the next chapter we consider completely positive matrices. We define Hadamard and Kronecker product and show that both of these products of completely positive matrices are completely positive. We introduce one method which enables us to verify whether the doubly nonnegative matrix is completely positive. We define the concept of a convex cone and show that the set of all completely positive matrices is a closed convex cone. With algebraic and geometric approach we show that small matrices are completely positive. At the end of the thesis we treat diagonally dominant matrices and show that nonnegative symmetric diagonally dominant matrices are completely positive. We also define a comparison matrix and show that the matrix A is completely positive if it is a symmetric nonnegative matrix and if its comparison matrix is positive semidefinite.