Condition Numbers of Gaussian Random Matrices

TL;DR

This paper derives probabilistic bounds and expectations for the condition number of Gaussian random matrices, providing insights into their numerical stability characteristics.

Contribution

It establishes new bounds on the distribution and expected logarithm of the condition number for real and complex Gaussian matrices.

Findings

Probability bounds depend on matrix dimensions and a universal constant.

Expected log condition number is bounded by a function of matrix dimensions.

Results are applicable to both real and complex Gaussian matrices.

Abstract

Let $G_{m \times n}$ be an $m \times n$ real random matrix whose elements are independent and identically distributed standard normal random variables, and let $\kappa_2(G_{m \times n})$ be the 2-norm condition number of $G_{m \times n}$. We prove that, for any $m \geq 2$, $n \geq 2$ and $x \geq |n-m|+1$, $\kappa_2(G_{m \times n})$ satisfies $ \frac{1}{\sqrt{2\pi}} ({c}/{x})^{|n-m|+1} < P(\frac{\kappa_2(G_{m \times n})} {{n}/{(|n-m|+1)}}> x) < \frac{1}{\sqrt{2\pi}} ({C}/{x})^{|n-m|+1}, $ where $0.245 \leq c \leq 2.000$ and $ 5.013 \leq C \leq 6.414$ are universal positive constants independent of $m$, $n$ and $x$. Moreover, for any $m \geq 2$ and $n \geq 2$, $ E(\log\kappa_2(G_{m \times n})) < \log \frac{n}{|n-m|+1} + 2.258. $ A similar pair of results for complex Gaussian random matrices is also established.