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

• Published in: IEEE transactions on information theory Vol. 68; no. 10; pp. 6716 - 6737
• Main Authors: Dissanayake, Pasan, Dharmawansa, Prathapasinghe, Chen, Yang
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.10.2022; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Asymptotic properties; Condition number; Correlation; Covariance matrices; Covariance matrix; cumulative distribution function (cdf); Eigenvalues; Eigenvalues and eigenfunctions; hypergeometric function of two matrix arguments; Instruments; Mathematical analysis; MIMO communication; moment generating function (mgf); orthogonal polynomials; Polynomials; probability density function (pdf); Probability density functions; Sensors; Signal detection; single-spiked covariance; Wishart matrix
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2022.3180286

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 single-spiked 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 scaled condition number 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>.