  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 $m = n$.
