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

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.