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.
Highlights
Let u ∈ Cn be a random vector uniformly distributed on the unit sphere
Eigenvectors and controllability rotationally invariant, and the individual eigenvectors of N have the same distribution as u above
In view of the universality phenomenon in random matrix theory, it is natural to conjecture that some of the properties that u possesses should hold for the eigenvectors of N
Summary
Let u ∈ Cn be a random vector uniformly distributed on the unit sphere. It follows that u has the same distribution as n i=1. Ξn are independent and identically distributed (iid) standard complex Gaussian random variables. Let N be a random matrix of size n × n whose entries are iid random variables. We quantify some of these properties of the eigenvectors for iid random matrices. Let W = (wij) be an n × n real symmetric random matrix whose entries wij, 1 ≤ i ≤ j ≤ n are iid copies of ξ. The goal of this work is to establish a version of Theorem 1.1 for non-Hermitian random matrices. We develop upon the techniques introduced by Ge [37] in order to overcome these difficulties
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
