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The Eigenvectors of Single-spiked Complex Wishart Matrices: Finite and Asymptotic Analyses
Prathapasinghe Dharmawansa1; Pasan Dissanayake1; Yang Chen2
2022-07-04
Source PublicationIEEE transactions on information theory
ISSN0018-9448
Abstract

Let W ∈ C n×n be a single-spiked Wishart matrix in the class W ∼ CWn(m, In + θvv† ) with m ≥ n, where In is the n × n identity matrix, v ∈ C n×1 is an arbitrary vector with unit Euclidean norm, θ ≥ 0 is a non-random parameter, and (·) † 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) ℓ = v †uℓ 2 ∈ (0, 1) for ℓ = 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 χ 2 2/2(1+θ), where χ 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.

KeywordConvergence In Distribution Eigenvalues Eigenvectors Gauss Hypergeometric Function Hypergeometric Function Of Two Matrix Arguments Laguerre Polynomials Moment Generating Function (M.g.f.) Probability Density Function (P.d.f.) Single-spiked Covariance Wishart Matrix
DOI10.1109/TIT.2022.3187919
PublisherIEEE
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Document TypeJournal article
CollectionFaculty of Science and Technology
DEPARTMENT OF MATHEMATICS
Affiliation1.University of Moratuwa, Moratuwa, Sri Lanka
2.Faculty of Science and Technology, University of Macau, Macau, P. R. China
Recommended Citation
GB/T 7714
Prathapasinghe Dharmawansa,Pasan Dissanayake,Yang Chen. The Eigenvectors of Single-spiked Complex Wishart Matrices: Finite and Asymptotic Analyses[J]. IEEE transactions on information theory,2022.
APA Prathapasinghe Dharmawansa,Pasan Dissanayake,&Yang Chen.(2022).The Eigenvectors of Single-spiked Complex Wishart Matrices: Finite and Asymptotic Analyses.IEEE transactions on information theory.
MLA Prathapasinghe Dharmawansa,et al."The Eigenvectors of Single-spiked Complex Wishart Matrices: Finite and Asymptotic Analyses".IEEE transactions on information theory (2022).
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