The Eigenvectors of Single-spiked Complex Wishart Matrices: Finite and Asymptotic Analyses

Let $\mathbf{W}\in\mathbb{C}^{n\times n}$ be a {\it single-spiked} Wishart matrix in the class $\mathbf{W}\sim \mathcal{CW}_n(m,\mathbf{I}_n+ \theta \mathbf{v}\mathbf{v}^\dagger)$ with $m\geq n$, where $\mathbf{I}_n$ is the $n\times n$ identity matrix, $\mathbf{v}\in\mathbb{C}^{n\times 1}$ 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 $\mathbf{u}_1$ and $\mathbf{u}_n$ denote the eigenvectors corresponding to the samllest and the largest eigenvalues of $\mathbf{W}$, respectively. This paper investigates the probability density function (p.d.f.) of the random quantity $Z_{\ell}^{(n)}=\left|\mathbf{v}^\dagger\mathbf{u}_\ell\right|^2\in(0,1)$ 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 $\mathbf{u}_1$ 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.