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This is a Quantitative Study study.

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October 16, 2025Open Access

The Determinantal Matroid

Key Points

Unique completability of partially observed matrices is central to matrix completion problems.
Criteria for detecting dependent sets in the algebraic matroid are thoroughly established.
A comprehensive description of matroid bases enhances understanding of determinantal structures.
The study links algebraic matroids to real-world applications in data reconstruction.

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.

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Cite This Study

Nicklasson et al. (Tue,) studied this question.

synapsesocial.com/papers/68f0d5eb105731330a2b221a https://doi.org/https://doi.org/10.48550/arxiv.2502.18222