Eigenvectors and controllability of non-Hermitian random matrices and directed graphs
Eigenvectors and controllability of non-Hermitian random matrices and directed graphs
复制标题
DOI：
10.1214/21-ejp588
复制
发表时间：
2020-04
期刊：
影响因子：
--
通讯作者：
K. Luh;Sean O’Rourke
中科院分区：
文献类型：
--
作者：
K. Luh;Sean O’Rourke
研究了具有iid元的随机矩阵的特征向量和特征值。设$N$是一个随机矩阵，其iid元素具有对称分布。对于每一个单位特征向量$\mathbf{v}$的$N$我们的主要结果提供了一个小球概率界的线性组合的坐标$\mathbf{v}$。我们的结果推广了Meehan和Nguyen以及Touri和第二作者关于随机对称矩阵的工作。沿着的方式，我们提供了一个最佳估计的概率，iid矩阵有简单的频谱，改进了最近的结果。我们的技术也使我们能够建立一个随机有向图的邻接矩阵的类似结果，作为一个应用程序，我们建立有向图的网络控制系统的可控性。
We study the eigenvectors and eigenvalues of random matrices with iid entries. Let $N$ be a random matrix with iid entries which have symmetric distribution. For each unit eigenvector $\mathbf{v}$ of $N$ our main results provide a small ball probability bound for linear combinations of the coordinates of $\mathbf{v}$. Our results generalize the works of Meehan and Nguyen as well as Touri and the second author for random symmetric matrices. Along the way, we provide an optimal estimate of the probability that an iid matrix has simple spectrum, improving a recent result of Ge. Our techniques also allow us to establish analogous results for the adjacency matrix of a random directed graph, and as an application we establish controllability properties of network control systems on directed graphs.