  Let $\mathbf{X}\in\mathbb{C}^{n\times m}$ ($m\geq n$) be a random matrix with
independent columns each distributed as complex multivariate Gaussian with zero
mean and {\it single-spiked} covariance matrix $\mathbf{I}_n+ \eta
\mathbf{u}\mathbf{u}^*$, where $\mathbf{I}_n$ is the $n\times n$ identity
matrix, {\color{blue}$\mathbf{u}\in\mathbb{C}^{n\times 1}$} is an arbitrary
vector with unit Euclidean norm, $\eta\geq 0$ is a non-random parameter, and
$(\cdot)^*$ represents the conjugate-transpose. This paper investigates the
distribution of the random quantity
$\kappa_{\text{SC}}^2(\mathbf{X})=\sum_{k=1}^n \lambda_k/\lambda_1$, where
{\color{blue}$0\le \lambda_1\le \lambda_2\le \ldots\leq \lambda_n<\infty$} are
the ordered eigenvalues of $\mathbf{X}\mathbf{X}^*$ (i.e., single-spiked
Wishart matrix). This random quantity is intimately related to the so called
{\it scaled condition number} or the Demmel condition number (i.e.,
$\kappa_{\text{SC}}(\mathbf{X})$) and the minimum eigenvalue of the fixed trace
Wishart-Laguerre ensemble (i.e., $\kappa_{\text{SC}}^{-2}(\mathbf{X})$). In
particular, we use an orthogonal polynomial approach to derive an exact
expression for the probability density function of
$\kappa_{\text{SC}}^2(\mathbf{X})$ which is amenable to asymptotic analysis as
matrix dimensions grow large. Our asymptotic results reveal that, as
$m,n\to\infty$ such that $m-n$ is fixed and when $\eta$ scales on the order of
$1/n$, $\kappa_{\text{SC}}^2(\mathbf{X})$ scales on the order of $n^3$. In this
respect we establish simple closed-form expressions for the limiting
distributions. {\color{blue}It turns out that, as $m,n\to\infty$ such that
$n/m\to c\in(0,1)$, properly centered $\kappa_{\text{SC}}^{2}(\mathbf{X})$
fluctuates on the scale $m^{\frac{1}{3}}$}.
