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 Affiliated with RC false Status 即將出版Forthcoming Distribution of the Scaled Condition Number of Single-spiked Complex Wishart Matrices Pasan Dissanayake1; Prathapasinghe Dharmawansa; Yang Chen2 2022-06-08 Source Publication IEEE Transactions on Information Theory ISSN 0018-9448 Volume 68Issue:10Pages:6716-6737 Abstract Let X ∈ Cn×m (m ≥ n) be a random matrix with independent columns each distributed as complex multivariate Gaussian with zero mean and single-spiked covariance matrix In + ηuu*, where In is the n × n identity matrix, u ∈ Cn×1 is an arbitrary vector with unit Euclidean norm, η ≥ 0 is a non-random parameter, and (∙)*. represents the conjugate-transpose. This paper investigates the distribution of the random quantity κ2SC(X) = Σnk=1 λk/λ1, where 0 ≤ λ1 ≤ λ2 ≤ . . . ≤ λn < ∞ are the ordered eigenvalues of XX* (i.e., single-spiked Wishart matrix). This random quantity is intimately related to the so called scaled condition number or the Demmel condition number (i.e., κSC(X)) and the minimum eigenvalue of the fixed trace Wishart-Laguerre ensemble (i.e., κ-2SC (X)). In particular, we use an orthogonal polynomial approach to derive an exact expression for the probability density function of κ2SC(X) which is amenable to asymptotic analysis as matrix dimensions grow large. Our asymptotic results reveal that, as m, n → ∞ such that m – n is fixed and when η scales on the order of 1/n, κ2SC(X) scales on the order of n3. In this respect we establish simple closed-form expressions for the limiting distributions. It turns out that, as m, n → ∞ such that n/m → c ∈ (0, 1), properly centered κ2SC(X) fluctuates on the scale m1/3. Keyword Condition Number Cumulative Distribution Function (C.d.f.) Eigenvalues Hypergeometric Function Of Two Matrix Arguments Moment Generating Function (M.g.f.) Orthogonal Polynomials Probability Density Function (P.d.f.) Single-spiked Covariance Wishart Matrix DOI 10.1109/TIT.2022.3180286 URL View the original Indexed By SCIE Language 英語English WOS Research Area Computer Science ; Engineering WOS Subject Computer Science, Information Systems ; Engineering, Electrical & Electronic WOS ID WOS:000854619600025 Publisher IEEE Scopus ID 2-s2.0-85131760570 Fulltext Access Citation statistics Cited Times [WOS]:0   [WOS Record]     [Related Records in WOS] Document Type Journal article Collection Faculty of Science and TechnologyDEPARTMENT OF MATHEMATICS Affiliation 1.University of Moratuwa, Moratuwa, Sri Lanka2.Faculty of Science and Technology, University of Macau, Macau, P. R. China Recommended CitationGB/T 7714 Pasan Dissanayake,Prathapasinghe Dharmawansa,Yang Chen. Distribution of the Scaled Condition Number of Single-spiked Complex Wishart Matrices[J]. IEEE Transactions on Information Theory,2022,68(10):6716-6737. APA Pasan Dissanayake,Prathapasinghe Dharmawansa,&Yang Chen.(2022).Distribution of the Scaled Condition Number of Single-spiked Complex Wishart Matrices.IEEE Transactions on Information Theory,68(10),6716-6737. MLA Pasan Dissanayake,et al."Distribution of the Scaled Condition Number of Single-spiked Complex Wishart Matrices".IEEE Transactions on Information Theory 68.10(2022):6716-6737.
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