Computational Resolution of Hadamard Product Factorization for $4 \times 4$ Matrices

TL;DR

This paper investigates the expressibility of 4x4 full-rank matrices as Hadamard products of two rank-2 matrices, revealing a significant number of counterexamples and uncovering low-dimensional algebraic structures using computational and machine learning methods.

Contribution

It computationally resolves an open problem by identifying counterexamples and demonstrating the low-dimensional structure of expressible matrices.

Findings

Identified 5,304 counterexamples among 20,160 binary matrices

Matrix density predicts expressibility with 95.7% accuracy

Expressible matrices form an approximately 10-dimensional variety

Abstract

We computationally resolve an open problem concerning the expressibility of $4 \times 4$ full-rank matrices as Hadamard products of two rank-2 matrices. Through exhaustive search over $\mathbb{F}_2$, we identify 5,304 counterexamples among the 20,160 full-rank binary matrices (26.3\%). We verify that these counterexamples remain valid over $\mathbb{Z}$ through sign enumeration and provide strong numerical evidence for their validity over $\mathbb{R}$. Remarkably, our analysis reveals that matrix density (number of ones) is highly predictive of expressibility, achieving 95.7\% classification accuracy. Using modern machine learning techniques, we discover that expressible matrices lie on an approximately 10-dimensional variety within the 16-dimensional ambient space, despite the naive parameter count of 24 (12 parameters each for two $4 \times 4$ rank-2 matrices). This emergent low-dimensional structure suggests deep algebraic constraints governing Hadamard factorizability.