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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.

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