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

• URL: http://arxiv.org/abs/2508.14901v1
• Date: Thu, 31 Jul 2025 21:00:28 GMT
• Title: Computational Resolution of Hadamard Product Factorization for $4 \times 4$ Matrices
• Authors: Igor Rivin,
• Abstract summary: We find that expressible $4 times 4$ full-rank matrices lie on an approximately 10-dimensional variety within the 16-dimensional ambient space.<n>This emergent low-dimensional structure suggests deep algebraic constraints governing Hadamard factorizability.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

Related papers

• Spectral Estimation with Free Decompression [47.81955761814048]
  We introduce a novel method of "free decompression" to estimate the spectrum of very large (impalpable) matrices.<n>Our method can be used to extrapolate from the empirical spectral densities of small submatrices to infer the eigenspectrum of extremely large (impalpable) matrices.
  arXiv  Detail & Related papers  (2025-06-13T17:49:25Z)
• Effective criteria for entanglement witnesses in small dimensions [0.0]
  We present an effective set of necessary and sufficient criteria for block-positivity of order $4$ sequences over $mathbbC$.<n>The method can be generalized to $mathcalHotimesmathcalH_d$ systems for $d>2$ to provide necessary but not sufficient criterion for block-positivity.
  arXiv  Detail & Related papers  (2025-06-10T23:14:52Z)
• Matrix encoding method in variational quantum singular value decomposition [49.494595696663524]
  We propose the variational quantum singular value decomposition based on encoding the elements of the considered $Ntimes N$ matrix into the state of a quantum system of appropriate dimension.<n> Controlled measurement is involved to avoid small success in ancilla measurement.
  arXiv  Detail & Related papers  (2025-03-19T07:01:38Z)
• High-Rank Irreducible Cartesian Tensor Decomposition and Bases of Equivariant Spaces [47.83974626445763]
  We construct path matrices for decomposition of Cartesian tensors up to rank $n=9$ with reduced and affordable complexity.<n>We prove and leverage that the concatenation of path matrices is an orthonormal change-of-basis matrix between the tensor product space and the spherical direct sum spaces.<n>We extend our result to the arbitrary tensor product and direct sum spaces, enabling free design between different spaces while keeping symmetry.
  arXiv  Detail & Related papers  (2024-12-24T08:25:38Z)
• A note on MDS Property of Circulant Matrices [3.069335774032178]
  In $2014$, Gupta and Ray proved that the circulant involutory matrices over the finite field $mathbbF_2m$ can not be maximum distance separable (MDS) This article delves into circulant matrices possessing these characteristics over the finite field $mathbbF_2m$.
  arXiv  Detail & Related papers  (2024-06-22T16:00:00Z)
• On MDS Property of g-Circulant Matrices [3.069335774032178]
  We first discuss $g$-circulant matrices with involutory and MDS properties. We then delve into $g$-circulant semi-involutory and semi-orthogonal matrices with entries from finite fields.
  arXiv  Detail & Related papers  (2024-06-22T15:18:31Z)
• A Systematic Construction Approach for All $4\times 4$ Involutory MDS Matrices [1.3332839594069594]
  We present several characterizations of involutory MDS matrices of even order. We propose a technique to systematically construct all $4 times 4$ involutory MDS matrices over a finite field.
  arXiv  Detail & Related papers  (2024-04-12T05:37:42Z)
• On the Counting of Involutory MDS Matrices [0.0]
  This paper enumerates Hadamard MDS and involutory Hadamard MDS matrices of order $4$ within the field $mathbbF_2r$. It also derives the count of Hadamard-MDS (NMDS) and involutory Hadamard NMDS matrices, each with exactly one zero in each row, of order $4$ over $mathbbF_2r$.
  arXiv  Detail & Related papers  (2023-09-29T18:57:00Z)
• One-sided Matrix Completion from Two Observations Per Row [95.87811229292056]
  We propose a natural algorithm that involves imputing the missing values of the matrix $XTX$. We evaluate our algorithm on one-sided recovery of synthetic data and low-coverage genome sequencing.
  arXiv  Detail & Related papers  (2023-06-06T22:35:16Z)
• On the well-spread property and its relation to linear regression [4.619541348328937]
  We show that consistent recovery of the parameter vector in a robust linear regression model is information-theoretically impossible. We show that it is possible to efficiently certify whether a given $n$-by-$d$ matrix is well-spread if the number of observations is quadratic in the ambient dimension.
  arXiv  Detail & Related papers  (2022-06-16T11:17:44Z)
• Spectral properties of sample covariance matrices arising from random matrices with independent non identically distributed columns [50.053491972003656]
  It was previously shown that the functionals $texttr(AR(z))$, for $R(z) = (frac1nXXT- zI_p)-1$ and $Ain mathcal M_p$ deterministic, have a standard deviation of order $O(|A|_* / sqrt n)$. Here, we show that $|mathbb E[R(z)] - tilde R(z)|_F
  arXiv  Detail & Related papers  (2021-09-06T14:21:43Z)
• Non-PSD Matrix Sketching with Applications to Regression and Optimization [56.730993511802865]
  We present dimensionality reduction methods for non-PSD and square-roots" matrices. We show how these techniques can be used for multiple downstream tasks.
  arXiv  Detail & Related papers  (2021-06-16T04:07:48Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.