AI 中文总结
该研究探讨离散随机矩阵$M_n(\xi)$为强非奇异的概率，证明其概率存在常数下界且增长因子有界，给出概率的紧渐近下界，并统计$n\leq9$的强非奇异二元矩阵精确数以推导伯努利情形的改进上界。
AI 中文摘要
我们研究离散随机矩阵$M_n(\xi)$为强非奇异的概率，强非奇异指其所有前主子矩阵均非奇异，该性质等价于存在LU分解。我们证明，对于支撑有限且$|\xi|_\infty<1$的任意离散随机变量$\xi$，$M_n(\xi)$为强非奇异的概率存在常数下界，且增长因子以$n^{5/2+\delta}$为界。此外，当$|\xi|_\infty\to0$时，我们给出该概率的紧渐近下界；最后，我们统计了$n\leq9$的强非奇异二元矩阵的精确数量，并据此推导伯努利情形下的改进上界。
英文摘要
We consider the probability that a discrete random matrix $M_n(ξ)$ 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 $ξ$ with finite support and $|ξ|_\infty < 1$, there is a constant probability that $M_n(ξ)$ is strongly non-singular with a growth factor bounded by $n^{5/2+δ}$. Furthermore, we provide a tight asymptotic lower bound for this probability as $|ξ|_\infty \to 0$. 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.
