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The Determinantal Matroid

Lisa Nicklasson, Manolis C. Tsakiris

TL;DR

The paper analyzes the algebraic matroid of the determinantal variety of $m\times n$ matrices with rank at most $r$, connecting to bounded-rank matrix completion. It introduces a general height-based dependence criterion, develops practical tools to detect dependent sets, and establishes a comprehensive framework for understanding unique completability, including equivalences between geometric fibers and field extensions. It also identifies structured base families, notably diagonal and anti-diagonal bases, and proves that non-intersecting anti-diagonal path families form bases while characterizing conditions under which such bases are uniquely completable. Collectively, these results advance identifiability and completion guarantees in rank-constrained matrix problems and illuminate the combinatorial structure of the determinantal matroid.

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.

The Determinantal Matroid

TL;DR

The paper analyzes the algebraic matroid of the determinantal variety of matrices with rank at most , connecting to bounded-rank matrix completion. It introduces a general height-based dependence criterion, develops practical tools to detect dependent sets, and establishes a comprehensive framework for understanding unique completability, including equivalences between geometric fibers and field extensions. It also identifies structured base families, notably diagonal and anti-diagonal bases, and proves that non-intersecting anti-diagonal path families form bases while characterizing conditions under which such bases are uniquely completable. Collectively, these results advance identifiability and completion guarantees in rank-constrained matrix problems and illuminate the combinatorial structure of the determinantal matroid.

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.

Paper Structure

This paper contains 10 sections, 26 theorems, 37 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries and overview
  3. The determinantal matroid and supports of linkage matching fields
  4. A general height criterion for dependence
  5. Detecting dependent sets
  6. Unique completability
  7. Generalities
  8. Dimension and codimension $1$
  9. Codimension $2$
  10. Diagonal and anti-diagonal bases

Key Result

Theorem 3.2

If $|\Omega| = r(m+n-r)$ and there is a partition $[n] = \bigcup_{\ell= 1}^r \mathcal{J}_\ell$ such that each $\Omega \cap ([m] \times \mathcal{J}_\ell)$ is a relaxed $(1,r,m)$-SLMF, then $\Omega$ is a base of the matroid $\mathcal{M}(r, [m] \times [n])$.

Figures (1)

  • Figure :

Theorems & Definitions (48)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 3.1: Tsakiris-AMS-2023
  • Theorem 3.2: Tsakiris-AMS-2023
  • Theorem 3.3: Tsakiris-AMS-2023
  • Example 3.4
  • Example 3.5
  • Theorem 4.1
  • ...and 38 more