Distribution of the Scaled Condition Number of Single-spiked Complex Wishart Matrices
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Let 𝐗∈ℂ^m× n (m≥ n) be a random matrix with independent rows each distributed as complex multivariate Gaussian with zero mean and single-spiked covariance matrix 𝐈_n+ η𝐮𝐮^*, where 𝐈_n is the n× n identity matrix, 𝐮∈ℂ^n× n is an arbitrary vector with a unit Euclidean norm, η≥ 0 is a non-random parameter, and (·)^* represents conjugate-transpose. This paper investigates the distribution of the random quantity κ_SC^2(𝐗)=∑_k=1^n λ_k/λ_1, where 0<λ_1<λ_2<…<λ_n<∞ are the ordered eigenvalues of 𝐗^*𝐗 (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., κ_SC(𝐗)) and the minimum eigenvalue of the fixed trace Wishart-Laguerre ensemble (i.e., κ_SC^-2(𝐗)). In particular, we use an orthogonal polynomial approach to derive an exact expression for the probability density function of κ_SC^2(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, κ_SC^2(𝐗) scales on the order of n^3. In this respect we establish simple closed-form expressions for the limiting distributions.
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