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Distribution of the Scaled Condition Number of Single-Spiked Complex Wishart Matrices.
- Source :
-
IEEE Transactions on Information Theory . Oct2022, Vol. 68 Issue 10, p6716-6737. 22p. - Publication Year :
- 2022
-
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 single-spiked covariance matrix $\mathbf {I}_{n}+ \eta \mathbf {u}\mathbf {u}^{*}$ , where $\mathbf {I}_{n}$ is the $n\times n$ identity matrix, $\mathbf {u}\in \mathbb {C}^{n\times 1}$ 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 $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 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. 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 $m^{\frac {1}{3}}$. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00189448
- Volume :
- 68
- Issue :
- 10
- Database :
- Academic Search Index
- Journal :
- IEEE Transactions on Information Theory
- Publication Type :
- Academic Journal
- Accession number :
- 159210747
- Full Text :
- https://doi.org/10.1109/TIT.2022.3180286