Completion Problems for Positive Matrices with Minimal Rank

Abstract

Matrix completion problems studies the partial matrices, that is, a rectangular matrix some of whose entries are specified, and the remainder entries are free variables of some indicated set. By a completion of a partial matrix we consider a specification of the free variables obtaining a conventional matrix. The basic type of these problems try to obtain conditions for the existence of a completion for a given partial matrix in a class of interest.In this work, we study the minimal rank completion problem when the partial matrix P has the specified entries equal to zero, and the remaining entries are positive real numbers. By a graph theoretic approach we introduce some approximations to the question. Furthermore, we obtain completions for some classes of positive pattern matrices with minimal rank.