Distributions of Demmel and Related Condition Numbers

TL;DR

This paper derives exact and asymptotic distributions for certain condition number metrics related to complex Gaussian matrices, extending Edelman's results and providing insights into their behavior as matrix dimensions grow large.

Contribution

It provides new exact formulas and asymptotic descriptions for the distributions of Demmel and related condition numbers for complex Gaussian matrices, generalizing prior work.

Findings

Exact probability density functions derived for the distributions.

Asymptotic densities scale on the order of n^3 as dimensions grow.

Closed-form expressions obtained for large matrix limits.

Abstract

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 $m = n$.