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

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