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 2 SC (X) = n k=1 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 −2 SC (X)). In particular, we use an orthogonal polynomial approach to derive an exact expression for the probability density function of κ 2 SC (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 2 SC (X) scales on the order of n 3 . In this respect we establish simple closed-form expressions for the limiting distributions.
Let W ∈ C n×n be a single-spiked Wishart matrix in the class W ∼ CW n (m, I n + θvv † ) with m ≥ n, where I n 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 u 1 and u n denote the eigenvectors corresponding to the samllest and the largest eigenvalues of W, respectively. This paper investigates the probability density function (p.d.f.) of the random quantityIn particular, we derive a finite dimensional closed-form p.d.f.forwhich 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 nZconverges 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 u 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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