Nonnegative and M- Matrices

Abstract

U ovome radu proučavat čemo posebnu kategoriju matrica. Definirat ćemo nenegativne matrice te uvesti pojam M–matrice. Navest ćemo primjere te karakterizaciju navedenih matrica. Promotrit ćemo nenegativne matrice čiji inverzi su M–matrice. Ako je A nenegativna matrica reda n i $A^{-1}$ je M–matrica, onda su gotovo glavni minori od A svakog reda nenegativni. Pokazat ćemo da je nesingularna matrica A reda p inverzna M–matrica ako i samo ako je $Q^{T}AQ + D$ inverzna matrica reda n, pri čemu je Q nenegativna matrica dimenzija p × n, s točno jednim pozitivnim elementom u svakom stupcu i D je pozitivna dijagonalna matrica. Ovo obuhvaća nekoliko činjenica o inverzima M–matrica kao posebnim slučajevima.This paper studies special matrix categories. We will define nonegative and introduce the term M–matrices. A characterization of a class of totally nonnegative matrices whose inverses are M–matrices is given. It is then shown that if A is nonnegative of order n and $A^{-1}$ is an M–matrix, then the almost principal minors of A of all orders are nonnegative. We show that a nonsingular p-by-p matrix A is an inverse M–matrix if and only if $Q^{T}AQ + D$ is an n-by-n inverse M–matrix whenever Q is a p-by-n nonnegative matrix with exactly one positive entry in each column and D is a positive diagonal matrix. This includes few facts about inverse M–matrices as special cases