The Determinant of $\{\pm 1\}$-Matrices and Oriented Hypergraphs

TL;DR

This paper introduces a novel method for calculating determinants of 1-matrices using oriented hypergraphs and cycle-covers, revealing new insights into their structure and connections to Hadamard's problem.

Contribution

It develops a hypergraph-based framework for determinant calculation, linking cycle-cover families to matrix properties and symmetries, and provides a way to recover matrix entries from cycle-cover signs.

Findings

Non-edge-monic families contribute zero to the Laplacian.

Edge-monic families sum to the absolute determinant value.

Identifies symmetries related to Hadamard's maximum determinant problem.

Abstract

The determinants of $\{\pm 1\}$-matrices are calculated by via the oriented hypergraphic Laplacian and summing over an incidence generalization of vertex cycle-covers. These cycle-covers are signed and partitioned into families based on their hyperedge containment. Every non-edge-monic family is shown to contribute a net value of $0$ to the Laplacian, while each edge-monic family is shown to sum to the absolute value of the determinant of the original incidence matrix. Simple symmetries are identified as well as their relationship to Hadamard's maximum determinant problem. Finally, the entries of the incidence matrix are reclaimed using only the signs of an adjacency-minimal set of cycle-covers from an edge-monic family.