标题： 对正交不变矩阵进行逐元素的非线性函数应用 显示英文标题

标题： Entrywise application of non-linear functions on orthogonally invariant matrices

摘要： 在本文中，我们研究对对称正交不变随机矩阵系进行逐元素应用非线性函数如何改变谱分布。 我们也处理多变量情况，其中对多个正交不变矩阵的元素应用多变量函数；甚至允许矩阵之间的相关性。 我们发现，在所有这些情况下，都存在高斯等价原理，即非线性函数的渐近效果与取所涉及矩阵的线性组合和一个额外的独立GOE相同。 在单个矩阵的情况下，ReLU函数和在两个矩阵的情况下，最大值函数提供了说明性的例子。 显示英文摘要

摘要： In this article, we investigate how the entrywise application of a non-linear function to symmetric orthogonally invariant random matrix ensembles alters the spectral distribution. We treat also the multivariate case where we apply multivariate functions to entries of several orthogonally invariant matrices; where even correlations between the matrices are allowed. We find that in all those cases a Gaussian equivalence principle holds, that is, the asymptotic effect of the non-linear function is the same as taking a linear combination of the involved matrices and an additional independent GOE. The ReLU-function in the case of one matrix and the max-function in the case of two matrices provide illustrative examples.