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

by   Prathapasinghe Dharmawansa, et al.

Let ๐–โˆˆโ„‚^nร— n be a single-spiked Wishart matrix in the class ๐–โˆผ๐’ž๐’ฒ_n(m,๐ˆ_n+ ฮธ๐ฏ๐ฏ^โ€ ) with mโ‰ฅ n, where ๐ˆ_n is the nร— n identity matrix, ๐ฏโˆˆโ„‚^nร— 1 is an arbitrary vector with unit Euclidean norm, ฮธโ‰ฅ 0 is a non-random parameter, and (ยท)^โ€  represents the conjugate-transpose operator. Let ๐ฎ_1 and ๐ฎ_n denote the eigenvectors corresponding to the samllest and the largest eigenvalues of ๐–, respectively. This paper investigates the probability density function (p.d.f.) of the random quantity Z_โ„“^(n)=|๐ฏ^โ€ ๐ฎ_โ„“|^2โˆˆ(0,1) for โ„“=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 ฯ‡^2_2/2(1+ฮธ), where ฯ‡_2^2 denotes a chi-squared random variable with two degrees of freedom. This reveals that ๐ฎ_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.


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