Abstract
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 v of N our main results provide a small ball probability bound for linear combinations of the coordinates of v. Our results generalize the works of Meehan and Nguyen [59] as well as Touri and the second author [67, 68, 69] 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 [37]. 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.
Acknowledgments
We thank Hoi H. Nguyen for pointing out reference [37]. The second author thanks Behrouz Touri for introducing him to the problem and answering numerous questions. Finally, we thank the anonymous referree for their careful reading of the manuscript and helpful comments.
Citation
Kyle Luh. Sean O’Rourke. "Eigenvectors and controllability of non-Hermitian random matrices and directed graphs." Electron. J. Probab. 26 1 - 43, 2021. https://doi.org/10.1214/21-EJP588
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