Distribution of the Scaled Condition Number of Single-spiked Complex Wishart Matrices

TL;DR

This paper derives the exact distribution and asymptotic behavior of the scaled condition number for single-spiked complex Wishart matrices, revealing its scaling properties as matrix dimensions grow large.

Contribution

It provides a novel orthogonal polynomial approach to exactly characterize the distribution of the scaled condition number in single-spiked Wishart matrices and analyzes its asymptotic behavior.

Findings

Exact probability density function derived for the scaled condition number.

Asymptotic scaling of the condition number as matrix dimensions grow large.

Limiting distributions established for large matrix sizes.

Abstract

Let $\mathbf{X}\in\mathbb{C}^{n\times m}$ ($m\geq n$) be a random matrix with independent columns each distributed as complex multivariate Gaussian with zero mean and {\it single-spiked} covariance matrix $\mathbf{I}_n+ \eta \mathbf{u}\mathbf{u}^*$, where $\mathbf{I}_n$ is the $n\times n$ identity matrix, {\color{blue}$\mathbf{u}\in\mathbb{C}^{n\times 1}$} is an arbitrary vector with unit Euclidean norm, $\eta\geq 0$ is a non-random parameter, and $(\cdot)^*$ represents the conjugate-transpose. This paper investigates the distribution of the random quantity $\kappa_{\text{SC}}^2(\mathbf{X})=\sum_{k=1}^n \lambda_k/\lambda_1$, where {\color{blue}$0\le \lambda_1\le \lambda_2\le \ldots\leq \lambda_n<\infty$} are the ordered eigenvalues of $\mathbf{X}\mathbf{X}^*$ (i.e., single-spiked Wishart matrix). This random quantity is intimately related to the so called {\it scaled condition number} or the Demmel condition number (i.e., $\kappa_{\text{SC}}(\mathbf{X})$) and the minimum eigenvalue of the fixed trace Wishart-Laguerre ensemble (i.e., $\kappa_{\text{SC}}^{-2}(\mathbf{X})$). In particular, we use an orthogonal polynomial approach to derive an exact expression for the probability density function of $\kappa_{\text{SC}}^2(\mathbf{X})$ which is amenable to asymptotic analysis as matrix dimensions grow large. Our asymptotic results reveal that, as $m,n\to\infty$ such that $m-n$ is fixed and when $\eta$ scales on the order of $1/n$, $\kappa_{\text{SC}}^2(\mathbf{X})$ scales on the order of $n^3$. In this respect we establish simple closed-form expressions for the limiting distributions. {\color{blue}It turns out that, as $m,n\to\infty$ such that $n/m\to c\in(0,1)$, properly centered $\kappa_{\text{SC}}^{2}(\mathbf{X})$ fluctuates on the scale $m^{\frac{1}{3}}$}.