Let X∈Cn×m\mathbf {X}\in \mathbb {C}^{n\times m} ( m≥nm\geq n ) be a random matrix with independent columns each distributed as complex multivariate Gaussian with zero mean and single-spiked covariance matrix In+ηuu∗\mathbf {I}_{n}+ \eta \mathbf {u}\mathbf {u}^{*} , where In\mathbf {I}_{n} is the n×nn\times n identity matrix, u∈Cn×1\mathbf {u}\in \mathbb {C}^{n\times 1} is an arbitrary vector with unit Euclidean norm, η≥0\eta \geq 0 is a non-random parameter, and (⋅)∗(\cdot)^{*} represents the conjugate-transpose. This paper investigates the distribution of the random quantity κ2SC(X)=∑nk=1λk/λ1\kappa _{\text {SC}}^{2}(\mathbf {X})=\sum _{k=1}^{n} \lambda _{k}/\lambda _{1} , where 0≤λ1≤λ2≤…≤λn<∞0\le \lambda _{1}\le \lambda _{2}\le \ldots \leq \lambda _{n} < \infty are the ordered eigenvalues of XX∗\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., κSC(X)\kappa _{\text {SC}}(\mathbf {X}) ) and the minimum eigenvalue of the fixed trace Wishart-Laguerre ensemble (i.e., κ−2SC(X)\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 κ2SC(X)\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→∞m,n\to \infty such that m−nm-n is fixed and when η\eta scales on the order of 1/n1/n , κ2SC(X)\kappa _{\text {SC}}^{2}(\mathbf {X}) scales on the order of n3n^{3} . In this respect we establish simple closed-form expressions for the limiting distributions. It turns out that, as m,n→∞m,n\to \infty such that n/m→c∈(0,1)n/m\to c\in (0,1) , properly centered κ2SC(X)\kappa _{\text {SC}}^{2}(\mathbf {X}) fluctuates on the scale m13m^{\frac {1}{3}} .

中文翻译：

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单尖峰复数 Wishart 矩阵的缩放条件数分布


设 X∈Cn×m\mathbf {X}\in \mathbb {C}^{n\times m} ( m≥nm\geq n ) 是一个具有独立列的随机矩阵，每个列均分布为具有零均值的复多元高斯分布，并且单尖峰协方差矩阵 In+ηuu∗\mathbf {I}_{n}+ \eta \mathbf {u}\mathbf {u}^{*} ，其中 In\mathbf {I}_{n} 是 n ×nn\times n 单位矩阵， u∈Cn×1\mathbf {u}\in \mathbb {C}^{n\times 1} 是单位欧氏范数的任意向量， η≥0\eta \geq 0 是非随机参数， (⋅)*(\cdot)^{*} 表示共轭转置。本文研究随机量 κ2SC(X)=Σnk=1λk/λ1\kappa _{\text {SC}}^{2}(\mathbf {X})=\sum _{k=1} 的分布^{n} \lambda _{k}/\lambda _{1} ，其中 0≤λ1≤λ2≤…≤λn<∞0\le \lambda _{1}\le \lambda _{2}\le \ ldots \leq \lambda _{n} < \infty 是 XX*\mathbf {X}\mathbf {X}^{*} 的有序特征值（即单尖峰 Wishart 矩阵）。该随机量与所谓的缩放条件数或戴梅尔条件数（即 κSC(X)\kappa _{\text {SC}}(\mathbf {X}) ）以及固定的最小特征值密切相关追踪 Wishart-Laguerre 系综（即 κ−2SC(X)\kappa _{\text {SC}}^{-2}(\mathbf {X}) ）。特别是，我们使用正交多项式方法来推导出概率密度函数 κ2SC(X)\kappa _{\text {SC}}^{2}(\mathbf {X}) 的精确表达式，该表达式适合渐近随着矩阵维度变大进行分析。我们的渐进结果表明，当 m,n→∞m,n\to \infty 使得 m−nm-n 固定时，当 η\eta 按 1/n1/n 的量级缩放时，κ2SC(X)\kappa _{\text {SC}}^{2}(\mathbf {X}) 按 n3n^{3} 的顺序缩放。在这方面，我们为极限分布建立了简单的封闭式表达式。 事实证明，当 m,n→∞m,n\to \infty 使得 n/m→c∈(0,1)n/m\to c\in (0,1) 时，正确居中 κ2SC(X )\kappa _{\text {SC}}^{2}(\mathbf {X}) 在 m13m^{\frac {1}{3}} 范围内波动。