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arXiv · 2608.08998

LU Factorization of Discrete Random Matrices

Samuel Orellana Mateo·John Urschel·Nicholas West
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Abstract

We consider the probability that a discrete random matrix $M_n(\xi)$ is \emph{strongly non-singular}, meaning all its leading principal submatrices are non-singular. This property is equivalent to the existence of an LU factorization. We show that for any discrete random variable $\xi$ with finite support and $|\xi|_\infty < 1$, there is a constant probability that $M_n(\xi)$ is strongly non-singular with a growth factor bounded by $n^{5/2+\delta}$. Furthermore, we provide a tight asymptotic lower bound for this probability as $|\xi|_\infty \to 0$. Finally, we provide exact counts for strongly non-singular binary matrices up to $n=9$ and use these to derive improved upper bounds for the Bernoulli case.

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Samuel Orellana Mateo, John Urschel, Nicholas West. 2026-08-10. LU Factorization of Discrete Random Matrices. https://arxiv.org/abs/2608.08998

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