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Distribution of the Scaled Condition Number of Single-spiked Complex Wishart Matrices

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 $\mathbf{I}_n+ η\mathbf{u}\mathbf{u}^*$, 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, $η\geq 0$ is a non-random parameter, and $(\cdot)^*$ represents the conjugate-transpose. This paper investigates the distribution of the random quantity $κ_{\text{SC}}^2(\mathbf{X})=\sum_{k=1}^n λ_k/λ_1$, where {\color{blue}$0\le λ_1\le λ_2\le \ldots\leq λ_n<\infty$} 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., $κ_{\text{SC}}(\mathbf{X})$) and the minimum eigenvalue of the fixed trace Wishart-Laguerre ensemble (i.e., $κ_{\text{SC}}^{-2}(\mathbf{X})$). In particular, we use an orthogonal polynomial approach to derive an exact expression for the probability density function of $κ_{\text{SC}}^2(\mathbf{X})$ 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 $η$ scales on the order of $1/n$, $κ_{\text{SC}}^2(\mathbf{X})$ 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 $κ_{\text{SC}}^{2}(\mathbf{X})$ fluctuates on the scale $m^{\frac{1}{3}}$}.