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Distribution of the Scaled Condition Number of Single-spiked Complex Wishart Matrices
Pasan Dissanayake1; Prathapasinghe Dharmawansa; Yang Chen2
Source PublicationIEEE Transactions on Information Theory

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.

KeywordCondition 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
URLView the original
Indexed BySCIE
WOS Research AreaComputer Science ; Engineering
WOS SubjectComputer Science, Information Systems ; Engineering, Electrical & Electronic
WOS IDWOS:000854619600025
Scopus ID2-s2.0-85131760570
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Document TypeJournal article
CollectionFaculty of Science and Technology
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
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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