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

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.

Overlaps, Eigenvalue Gaps, and Pseudospectrum under real Ginibre and Absolutely Continuous Perturbations

“…for every > 0 and every > > − . In very recent work, Luh and O'Rourke [LO20] build on Ge's result, dropping the mean zero assumption and extending the range of all the way down to 0:…”

“…A secondary contribution of the paper, which also plays a role in the proof above, is a polynomial (in / ) lower bound on the minimum eigenvalue gap: gap( ) ∶= min ≠ | − | which holds with high probability (Theorem 1.6). The novelty of this result in comparison to existing minimum gap bounds (such as [Ge17,LO20]) is that it works for heterogeneous non-centered random matrices , as opposed to only matrices with i.i.d. entries.…”

Small Ball Probability of the Least Singular Value Ofnormalized Random Matrices: Universal Lower Bounds

“…(The analogous problem for i.i.d. random matrices has also been solved, see [16] and [23]). The difference between these two problems is as follows: the problem of distinct eigenvalues only requires singular value estimates at one location, whereas Conjecture (1.6) requires singular value estimates at different locations.…”

Recent progress in combinatorial random matrix theory

“…In this direction, Ge [40] proved the analogue of Theorem 8.3, showing that with probability 1 − o(1), the spectrum of M n is simple. In a very recent paper, Luh and O'rourke [54] proved the first exponential bound, showing that the probability that the spectrum of M n is not simple is at most 2 −cn , for some constant c > 0. It looks plausible that Conjecture 8.4 holds for M n as well.…”