Distribution of the scaled condition number of single-spiked complex wishart matrices

Abstract

Let X 2 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 + uu , where In is the n n identity matrix, u 2 Cn 1 is an arbitrary vector with unit Euclidean norm, 0 is a nonrandom parameter, and ( ) represents the conjugate-transpose. This paper investigates the distribution of the random quantity 2 SC(X) = Pn k=1 k= 1, where 0 1 2 : : : n < 1 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., SC(X)) and the minimum eigenvalue of the fixed trace Wishart-Laguerre ensemble (i.e., ð€€€2 SC (X)). In particular, we use an orthogonal polynomial approach to derive an exact expression for the probability density function of 2 SC(X) which is amenable to asymptotic analysis as matrix dimensions grow large. Our asymptotic results reveal that, as m; n ! 1 such that m ð€€€ n is fixed and when scales on the order of 1=n, 2 SC(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 ! 1 such that n=m ! c 2 (0; 1), properly centered 2 SC(X) fluctuates on the scale m 1 3 .