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

Abstract

Let W 2 Cn n be a single-spiked Wishart matrix in the class W CWn(m; In + vvy) with m n, where In is the n n identity matrix, v 2 Cn 1 is an arbitrary vector with unit Euclidean norm, 0 is a non-random parameter, and ( )y represents the conjugate-transpose operator. Let u1 and un 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(n) ` =

vyu`

2 2 (0; 1) for ` = 1; n. In particular, we derive a finite dimensional closed-form p.d.f. for Z(n) 1 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(n) 1 converges in distribution to 2 2 =2(1 + ), where 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.