Eigenvectors and controllability of non-Hermitian random matrices and directed graphs

We study the eigenvectors and eigenvalues of random matrices with iid entries. Let N be a matrix entries which have symmetric distribution. For each unit eigenvector v our main results provide small ball probability bound for linear combinations coordinates v. Our generalize works Meehan Nguyen [59] as well Touri second author [67, 68, 69] matrices. Along way, we an optimal estimate that has simple spectrum, improving recent result Ge [37]. techniques also allow us to establish analogous adjacency directed graph, application controllability properties network control systems on graphs.