高斯扰动的矩阵正则化效应
Matrix regularizing effects of Gaussian perturbations
摘要：
噪声的加入对埃尔米特矩阵具有正则化效应。在此研究了$H=A+V$的这种效应，其中$A$是基础矩阵，而$V$是从GOE或GUE随机矩阵集合中采样的。我们绑定的$H$在一个区间内的特征值的平均数，并目前的Frobenius和运营商规范的分布的$H^{-1}$和规范的分布的$H^{-1}$适用于一个固定的向量的尾界。在$A$中，边界是一致的，并且超出实际上界不超过乘法常数。在一个区间的多个特征值的概率也估计。
英文摘要：
The addition of noise has a regularizing effect on Hermitian matrices. This effect is studied here for $H=A+V$, where $A$ is the base matrix and $V$ is sampled from the GOE or the GUE random matrix ensembles. We bound the mean number of eigenvalues of $H$ in an interval, and present tail bounds for the distribution of the Frobenius and operator norms of $H^{-1}$ and for the distribution of the norm of $H^{-1}$ applied to a fixed vector. The bounds are uniform in $A$ and exceed the actual suprema by no more than multiplicative constants. The probability of multiple eigenvalues in an interval is also estimated.
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