The totally positive completion problem

The General Totally Positive Matrix Completion Problem with Few Unspecified Entries

For m by n partial totally positive matrices with exactly one unspeci ed entry the set of positions for that entry that guarantee completability to a totally positive matrix are characterized They are the positions i j i j and the positions i j i j m n In each case the set of completing entries is an open and in nite in case i j or i m j n interval In the process some new structural results abo...

The Positive Definite Matrix Completion Problem: an Optimization Viewpoint∗

We look at the real positive (semi)definite matrix completion problem from the relative entropy minimization viewpoint. After the problem is transformed into the standard maxdet from, conditions are sought for existence of positive (semi)definite completions. Using basic tools of convex analysis results previously established using graph-theoretic or functional-analytic techniques are recovered...

Conditions for a totally positive completion in the case of a symmetrically placed cycle

Ela Conditions for a Totally Positive Completion in the Case of a Symmetrically Placed Cycle

Abstract. In earlier work, the labelled graphs G for which every combinatorially symmetric totally nonnegative matrix, the graph of whose specified entries is G, has a totally nonnegative completion were identified. For other graphs, additional conditions on the specified data must hold. Here, necessary and sufficient conditions on the specified data, when G is a cycle, are given for both the t...

The Tropical Totally Positive Grassmannian

Tropical algebraic geometry is the geometry of the tropical semiring (R,min,+). The theory of total positivity is a natural generalization of the study of matrices with all minors positive. In this paper we introduce the totally positive part of the tropicalization of an arbitrary affine variety, an object which has the structure of a polyhedral fan. We then investigate the case of the Grassman...