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Let $\mathrm {W}\in \mathbb {C}^{n\times n}$ be a single-spiked Wishart matrix in the class $\mathrm {W}\sim \mathcal {CW}_{n}(m,\mathrm {I}_{n}+ \theta \mathrm {v}\mathrm {v}^{\dagger}) $ with $m\geq n$ , where ${\mathrm {I}}_{n}$ is the $n\times n$ identity matrix, $\mathrm {v}\in \mathbb {C}^{n\times 1}$ is an arbitrary vector with unit Euclidean norm, $\theta \geq 0$ is a non-random parameter, and $(\cdot)^{\dagger} $ represents the conjugate-transpose operator. Let u1 and ${\mathrm {u}}_{n}$ denote the eigenvectors corresponding to the smallest and the largest eigenvalues of W, respectively. This paper investigates the probability density function (p.d.f.) of the random quantity $Z_{\ell }^{(n)}=\left |{\mathrm {v}^{\dagger} \mathrm {u}_\ell }\right |^{2}\in (0,1)$ for $\ell =1,n$. In particular, we derive a finite dimensional closed-form p.d.f. for $Z_{1}^{(n)}$ which is amenable to asymptotic analysis as $m,n$ diverges with $m-n$ fixed. It turns out that, in this asymptotic regime, the scaled random variable $nZ_{1}^{(n)}$ converges in distribution to $\chi ^{2}_{2}/2(1+\theta)$ , where $\chi _{2}^{2}$ denotes a chi-squared random variable with two degrees of freedom. This reveals that u1 can be used to infer information about the spike. On the other hand, the finite dimensional p.d.f. of $Z_{n}^{(n)}$ is expressed as a double integral in which the integrand contains a determinant of a square matrix of dimension $(n-2)$. Although a simple solution to this double integral seems intractable, for special configurations of $n=2,3$ , and 4, we obtain closed-form expressions. [ABSTRACT FROM AUTHOR]

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 single-spiked covariance matrix $\mathbf {I}_{n}+ \eta \mathbf {u}\mathbf {u}^{*}$ , where $\mathbf {I}_{n}$ is the $n\times n$ identity matrix, $\mathbf {u}\in \mathbb {C}^{n\times 1}$ 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 $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 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. 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 $m^{\frac {1}{3}}$. [ABSTRACT FROM AUTHOR]