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

Understand

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.