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摘要翻译：
本文提出了一个研究一般对称矩阵{bf H}0$的连续特征向量构成的子空间的稳定性的一般框架，当附加小扰动时，该子空间的稳定性是由一般对称矩阵{bf H}0$的连续特征向量构成的。这个问题在各种情况下都有关联，包括量子耗散(${\bf H}0$是哈密顿量）和金融风险控制（在这种情况下${\bf H}0$是资产收益协方差矩阵）。我们认为这个问题可以用重叠矩阵的奇异值来表述，这允许我们定义一个“保真度”距离。我们将我们的结果专门用于高斯正交${\bf H}0$的情形，对于这种情形，奇异值的全谱可以显式地计算出来。我们还考虑了当${\bf H}_0$是协方差矩阵时的情形，并用金融数据说明了我们的结果的有用性。对于顶部特征值比其它特征值大得多的特殊情况，可以进行详细的研究。特别是顶部特征向量与其真实方向所构成的角度的动力学定义了一类有趣的新随机过程。
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英文标题：
《Eigenvector dynamics: general theory and some applications》
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作者：
Romain Allez and Jean-Philippe Bouchaud
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最新提交年份：
2012
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分类信息：
一级分类：Physics 物理学
二级分类：Statistical Mechanics 统计力学
分类描述：Phase transitions, thermodynamics, field theory, non-equilibrium phenomena, renormalization group and scaling, integrable models, turbulence
相变，热力学，场论，非平衡现象，重整化群和标度，可积模型，湍流
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一级分类：Mathematics 数学
二级分类：Probability 概率
分类描述：Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
概率论与随机过程的理论与应用：例如中心极限定理，大偏差，随机微分方程，统计力学模型，排队论
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一级分类：Quantitative Finance 数量金融学
二级分类：Risk Management 风险管理
分类描述：Measurement and management of financial risks in trading, banking, insurance, corporate and other applications
衡量和管理贸易、银行、保险、企业和其他应用中的金融风险
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一级分类：Quantitative Finance 数量金融学
二级分类：Statistical Finance 统计金融
分类描述：Statistical, econometric and econophysics analyses with applications to financial markets and economic data
统计、计量经济学和经济物理学分析及其在金融市场和经济数据中的应用
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英文摘要：
We propose a general framework to study the stability of the subspace spanned by $P$ consecutive eigenvectors of a generic symmetric matrix ${\bf H}_0$, when a small perturbation is added. This problem is relevant in various contexts, including quantum dissipation (${\bf H}_0$ is then the Hamiltonian) and financial risk control (in which case ${\bf H}_0$ is the assets return covariance matrix). We argue that the problem can be formulated in terms of the singular values of an overlap matrix, that allows one to define a "fidelity" distance. We specialize our results for the case of a Gaussian Orthogonal ${\bf H}_0$, for which the full spectrum of singular values can be explicitly computed. We also consider the case when ${\bf H}_0$ is a covariance matrix and illustrate the usefulness of our results using financial data. The special case where the top eigenvalue is much larger than all the other ones can be investigated in full detail. In particular, the dynamics of the angle made by the top eigenvector and its true direction defines an interesting new class of random processes.
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PDF链接：
https://arxiv.org/pdf/1203.6228
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