The Determinantal Matroid

TL;DR

This paper explores the algebraic matroid structure related to matrix completion problems, providing criteria for dependence, bases, and conditions for unique matrix completion.

Contribution

It introduces new criteria for dependency, characterizes bases, and investigates unique completability within the algebraic matroid induced by matrix minors.

Findings

Criteria for detecting dependent sets in the matroid

Description of a family of bases of the matroid

Analysis of conditions for unique matrix completion

Abstract

We study the algebraic matroid induced by the ideal of (r+1)-minors of a matrix of variables over a field. This is inherently connected to the bounded-rank matrix completion problem, in which the aim is to complete a partially observed rank r matrix. We give criteria that detect dependent sets in the matroid, we describe a family of bases of the matroid, and we study the question of unique completability.