Computational Resolution of Hadamard Product Factorization for $4 \times 4$ Matrices
This resolves an open problem in linear algebra for researchers, revealing deep algebraic constraints on matrix factorization.
The paper tackled the problem of expressing 4x4 full-rank matrices as Hadamard products of two rank-2 matrices, finding 5,304 counterexamples among 20,160 binary matrices (26.3%) and showing that matrix density predicts expressibility with 95.7% accuracy.
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.