论文标题
非官员随机矩阵和有向图的特征向量和可控性
Eigenvectors and controllability of non-Hermitian random matrices and directed graphs
论文作者
论文摘要
我们研究带有IID条目的随机矩阵的特征值和特征值。令$ n $为带有对称分布的IID条目的随机矩阵。对于$ n $的每个单元eigenVector $ \ mathbf {v} $，我们的主要结果为$ \ mathbf {v} $的线性组合提供了一个小球概率。我们的结果概括了Meehan和Nguyen以及Touri的作品，以及随机对称矩阵的第二位作者。一路上，我们提供了IID矩阵具有简单频谱的可能性的最佳估计，从而改善了GE的最新结果。我们的技术还使我们能够为随机有向图的邻接矩阵建立类似的结果，作为应用程序，我们在有向图上建立了网络控制系统的可控性能。
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.
