发表机构
University of California, San Diego; University of California, Berkeley（加州大学圣地亚哥分校; 加州大学伯克利分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究随机矩阵在部分主元高斯消去中诱导的置换，分类蝴蝶矩阵的置换结构，分析GOE、GUE等模型的非均匀性及渐近行为，并揭示LAPACK实现细节对主元概率的影响，建立随机矩阵与置换组合学的新联系。
AI 中文摘要
部分主元高斯消去法（GEPP）仍然是求解稠密线性方程组 $A \mathbf x = \mathbf b$（其中 $A \in \mathbb C^{n\times n}$）最广泛使用的求解器。我们研究在GEPP分解 $PA = LU$ 中出现的置换 $\pi = \pi(A)$，该置换由置换矩阵因子 $P = P_\pi$ 编码。当输入矩阵是随机时，$\pi$ 也是随机的。对于大小为 $2^n$ 的随机标量蝴蝶矩阵（一个递归定义、最初为完全消除主元需求而引入的矩阵族），我们给出精确的GEPP分解，并将诱导置换完全分类为 $S_{2^n}$ 的一个 $2$-Sylow 子群中的元素，该子群包含于可分置换中。此外，均匀角模型在该子群上诱导均匀分布。对于GOE、GUE和独立同分布伯努利模型，当 $n \ge 2$ 时，诱导置换从不精确均匀。我们给出了GOE和GUE在首主元处偏离均匀性的精确速率，并给出证据表明这种非均匀性在置换子（permuton）意义上渐近消失。相比之下，对于次线性带宽的带状随机矩阵，包括三对角 $\beta$-Hermite 系综，诱导置换收敛到对角置换子。我们进一步表明，所得主元概率对实现选择敏感：标准LAPACK例程使用 $\ell^1$ 范数而非 $\ell^2$ 范数比较复数主元候选，将GUE(2)主元概率从 $1/\sqrt3$ 改变为 $2/3$。这些结果共同通过数值线性代数建立了随机矩阵理论与置换组合学之间的新联系。
英文摘要
Gaussian elimination with partial pivoting (GEPP) remains the most widely used solver for dense linear systems $A \mathbf x = \mathbf b$ for $A \in \mathbb C^{n\times n}$. We study the permutation $π= π(A)$ that arises in the GEPP factorization $PA = LU$, encoded by the permutation matrix factor $P = P_π$. When the input matrix is random, so is $π$. For random scalar butterfly matrices of size $2^n$ (a recursively defined family originally introduced to eliminate the need for pivoting altogether), we give the exact GEPP factorization and fully classify the induced permutation as an element of a $2$-Sylow subgroup of $S_{2^n}$ contained in the separable permutations. Moreover, the uniform-angle model induces the uniform distribution on this subgroup. For the GOE, GUE, and iid Bernoulli models, the induced permutation is never exactly uniform for $n \ge 2$. We give the precise rate of departure from uniformity at the leading pivot for the GOE and GUE, and give evidence that this non-uniformity vanishes asymptotically in the permuton sense. In contrast, for banded random matrices of sublinear bandwidth, including the tridiagonal $β$-Hermite ensembles, the induced permutation converges to the diagonal permuton. We further show that the resulting pivot probabilities are sensitive to implementation choices: standard LAPACK routines compare complex pivot candidates using the $\ell^1$ rather than $\ell^2$ norm, changing the GUE(2) pivot probability from $1/\sqrt3$ to $2/3$. Together these results establish a new connection between random matrix theory and permutation combinatorics through numerical linear algebra.
CommentsSupersedes arXiv:2506.20470 and incorporates part of arXiv:2410.20952, with substantially expanded results. First of three articles that together will supersede arXiv:2410.20952
