标题： Demmel及相关条件数的分布 显示英文标题

标题： Distributions of Demmel and Related Condition Numbers

摘要： 考虑一个随机矩阵$\mathbf{A}\in\mathbb{C}^{m\times n}$($m \geq n$)，其包含独立的零均值、单位方差的复高斯分布元素，令$0<\lambda_1\leq \lambda_{2}\leq ...\leq \lambda_n<\infty$表示$\mathbf{A}^{*}\mathbf{A}$的特征值，其中$(\cdot)^*$表示共轭转置。 本文研究了随机变量$\frac{\sum_{j=1}^n \lambda_j}{\lambda_k}$在$k = 1$和$k = 2$条件下的分布。 这两个变量与某些条件数度量有关，包括所谓的 Demmel 条件数，已知它在各种应用中出现。 对于这两种情况，我们推导出概率密度的新精确表达式，并研究矩阵维数增大时的渐近行为。 特别是，当$n$和$m$趋于无穷大且它们的差固定时，证明了两种密度都按$n^3$的数量级缩放。 经过适当的变换后，我们得到了渐近密度的确切表达式，在某些情况下得到了简单的闭合形式表达式。 我们的结果推广了 Edelman 关于$m = n$情况下 Demmel 条件数的工作。 显示英文摘要

摘要： Consider a random matrix $\mathbf{A}\in\mathbb{C}^{m\times n}$ ($m \geq n$) containing independent complex Gaussian entries with zero mean and unit variance, and let $0<\lambda_1\leq \lambda_{2}\leq ...\leq \lambda_n<\infty$ denote the eigenvalues of $\mathbf{A}^{*}\mathbf{A}$ where $(\cdot)^*$ represents conjugate-transpose. This paper investigates the distribution of the random variables $\frac{\sum_{j=1}^n \lambda_j}{\lambda_k}$, for $k = 1$ and $k = 2$. These two variables are related to certain condition number metrics, including the so-called Demmel condition number, which have been shown to arise in a variety of applications. For both cases, we derive new exact expressions for the probability densities, and establish the asymptotic behavior as the matrix dimensions grow large. In particular, it is shown that as $n$ and $m$ tend to infinity with their difference fixed, both densities scale on the order of $n^3$. After suitable transformations, we establish exact expressions for the asymptotic densities, obtaining simple closed-form expressions in some cases. Our results generalize the work of Edelman on the Demmel condition number for the case $m = n$.