Distributions of Demmel and Related Condition Numbers

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 $m = n$.

💡 Research Summary

The paper investigates the statistical behavior of two ratios built from the eigenvalues of a complex Wishart matrix, which are directly related to the Demmel condition number and a closely related metric. Let (\mathbf{A}\in\mathbb{C}^{m\times n}) be a random matrix with independent, zero‑mean, unit‑variance complex Gaussian entries and assume (m\ge n). The eigenvalues of the Hermitian positive‑definite matrix (\mathbf{A}^{*}\mathbf{A}) are denoted by (0<\lambda_{1}\le\lambda_{2}\le\cdots\le\lambda_{n}<\infty). The two random variables of interest are

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