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

Let $\mathbf{X}\in {\mathbb{C}}^{n\times m}$ ($m\ge n$) 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}$_n$isthe$n\times n$ identity matrix, {\color{blue}$\mathbf{u}\in {\mathbb{C}}^{n\times 1}$} is an arbitrary vector with unit Euclidean norm, $\eta \ge 0$ is a non-random parameter, and $(\cdot {)}^{\ast }$ represents the conjugate-transpose. This paper investigates the distribution of the random quantity ${\kappa }_{\text{SC}}^2(\mathbf{X})={\sum }_{k=1}^n{\lambda }_k\mathrm{/}{\lambda }_1$, where {\color{blue}$0\le {\lambda }_1\le {\lambda }_2\le \dots \le {\lambda }_n<\mathrm{\infty }$} are the ordered eigenvalues of $\mathbf{X}{\mathbf{X}}^{\ast }$ (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 $m,n\to \mathrm{\infty }$ such that $m-n$ is fixed and when $\eta$ scales on the order of $1\mathrm{/}n$, ${\kappa }_{\text{SC}}^2(\mathbf{X})$ 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 $m,n\to \mathrm{\infty }$ such that $n\mathrm{/}m\to c\in (0,1)$, properly centered ${\kappa }_{\text{SC}}^2(\mathbf{X})$ fluctuates on the scale $m^{\frac{1}{3}}$}.