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 $$\mathbf{W}$\sim $\mathcal{C}\mathcal{W}$_n(m,$\mathbf{I}$_n+ \theta $\mathbf{v}$$\mathbf{v}$^\dagger) $with$m\geq n$,where$$\mathbf{I}$_n$isthe$n\times n$identitymatrix,$$\mathbf{v}$\in$\mathbb{C}$^{n\times 1}$ is an arbitrary vector with unit Euclidean norm, $\theta \ge 0$ is a non-random parameter, and $(\cdot {)}^{†}$ 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 $Z_{\mathrm{\ell }}^{(n)}={\mid {\mathbf{v}}^{†}{\mathbf{u}}_{\mathrm{\ell }}\mid }^2\in (0,1)$ for $\mathrm{\ell }=1,n$. 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 $n{Z}_1^{(n)}$ converges in distribution to ${\chi }_2^2\mathrm{/}2(1+\theta )$, 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.