# On delocalization of eigenvectors of random non-Hermitian matrices

@article{Lytova2018OnDO, title={On delocalization of eigenvectors of random non-Hermitian matrices}, author={Anna Lytova and Konstantin E. Tikhomirov}, journal={Probability Theory and Related Fields}, year={2018}, volume={177}, pages={465-524} }

We study delocalization of null vectors and eigenvectors of random matrices with i.i.d entries. Let A be an $$n\times n$$ n × n random matrix with i.i.d real subgaussian entries of zero mean and unit variance. We show that with probability at least $$1-e^{-\log ^{2} n}$$ 1 - e - log 2 n $$\begin{aligned} \min \limits _{I\subset [n],\,|I|= m}\Vert \mathbf{{v}}_I\Vert \ge \frac{m^{3/2}}{n^{3/2}\log ^Cn}\Vert \mathbf{{v}}\Vert \end{aligned}$$ min I ⊂ [ n ] , | I | = m ‖ v I ‖ ≥ m 3 / 2 n 3 / 2 log… Expand

#### 12 Citations

Singularity of sparse Bernoulli matrices

- Mathematics
- 2020

Let $M_n$ be an $n\times n$ random matrix with i.i.d. Bernoulli(p) entries. We show that there is a universal constant $C\geq 1$ such that, whenever $p$ and $n$ satisfy $C\log n/n\leq p\leq C^{-1}$,… Expand

The Lower Bound for Koldobsky’s Slicing Inequality via Random Rounding

- Mathematics
- 2020

We study the lower bound for Koldobsky’s slicing inequality. We show that there exists a measure μ and a symmetric convex body \(K \subseteq \mathbb R^n\), such that for all \(\xi \in {{\mathbb… Expand

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

- Mathematics
- 2020

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… Expand

Spectral radius of random matrices with independent entries

- Mathematics, Physics
- 2019

We consider random $n\times n$ matrices $X$ with independent and centered entries and a general variance profile. We show that the spectral radius of $X$ converges with very high probability to the… Expand

Eigenvector delocalization for non-Hermitian random matrices and applications

- Mathematics, Computer Science
- Random Struct. Algorithms
- 2020

This work establishes delocalization bounds for eigenvectors of independent-entry random matrices and shows that with high probability every eigenvector is delocalized, meaning any subset of its coordinates carries an appropriate proportion of its mass. Expand

THE SMALLEST SINGULAR VALUE OF INHOMOGENEOUS SQUARE RANDOM MATRICES BY GALYNA

- 2020

This extends earlier results [27, 24] by removing the assumption of mean zero and identical distribution of the entries across the matrix, as well as the recent result [19] where the matrix was… Expand

The smallest singular value of heavy-tailed not necessarily i.i.d. random matrices via random rounding

- Mathematics
- 2018

We are concerned with the small ball behavior of the smallest singular value of random matrices. Often, establishing such results involves, in some capacity, a discretization of the unit sphere. This… Expand

Asymmetry Helps: Eigenvalue and Eigenvector Analyses of Asymmetrically Perturbed Low-Rank Matrices

- Computer Science, Medicine
- Annals of statistics
- 2021

It is demonstrated that the leading eigenvalue of the data matrix M can be O ( n ) times more accurate than its (unadjusted) leading singular value of M in eigen value estimation, and the eigen-decomposition approach is fully adaptive to heteroscedasticity of noise, without the need of any prior knowledge about the noise distributions. Expand

Inference for linear forms of eigenvectors under minimal eigenvalue separation: Asymmetry and heteroscedasticity

- Mathematics, Computer Science
- ArXiv
- 2020

This work develops algorithms that produce confidence intervals for linear forms of individual eigenvectors, based on eigen-decomposition of the asymmetric data matrix followed by a careful de-biasing scheme, and establishes procedures to construct optimalconfidence intervals for the eigenvalues of interest. Expand

Local elliptic law

- Mathematics, Physics
- 2021

The empirical eigenvalue distribution of the elliptic random matrix ensemble tends to the uniform measure on an ellipse in the complex plane as its dimension tends to infinity. We show this… Expand

#### References

SHOWING 1-10 OF 78 REFERENCES

Invertibility via distance for noncentered random matrices with continuous distributions

- Mathematics, Computer Science
- Random Struct. Algorithms
- 2020

The method is principally different from a standard approach involving a decomposition of the unit sphere and coverings, as well as an approach of Sankar-Spielman-Teng for non-centered Gaussian matrices. Expand

Random Band Matrices in the Delocalized Phase I: Quantum Unique Ergodicity and Universality

- Mathematics, Physics
- Communications on Pure and Applied Mathematics
- 2020

Consider $N\times N$ symmetric one-dimensional random band matrices with general distribution of the entries and band width $W \geq N^{3/4+\varepsilon}$ for any $\varepsilon>0$.
In the bulk of the… Expand

Universality of covariance matrices

- Mathematics
- 2011

In this paper we prove the universality of covariance matrices of the form $H_{N\times N}={X}^{\dagger}X$ where $X$ is an ${M\times N}$ rectangular matrix with independent real valued entries… Expand

The smallest singular value of random rectangular matrices with no moment assumptions on entries

- Mathematics
- 2014

Let δ > 1 and β > 0 be some real numbers. We prove that there are positive u, v, N0 depending only on β and δ with the following property: for any N,n such that N ≥ max(N0, δn), any N × n random… Expand

Bounding the smallest singular value of a random matrix without concentration

- Mathematics
- 2013

Given $X$ a random vector in ${\mathbb{R}}^n$, set $X_1,...,X_N$ to be independent copies of $X$ and let $\Gamma=\frac{1}{\sqrt{N}}\sum_{i=1}^N e_i$ be the matrix whose rows are… Expand

Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices

- Physics, Mathematics
- 2009

We consider $N\times N$ Hermitian random matrices with i.i.d. entries. The matrix
is normalized so that the average spacing between consecutive eigenvalues is of order
$1/N$. We study the… Expand

Delocalization and Diffusion Profile for Random Band Matrices

- Physics, Mathematics
- 2012

We consider Hermitian and symmetric random band matrices H = (hxy) in $${d\,\geqslant\,1}$$d⩾1 dimensions. The matrix entries hxy, indexed by $${x,y \in (\mathbb{Z}/L\mathbb{Z})^d}$$x,y∈(Z/LZ)d, are… Expand

Sharp lower bounds on the least singular value of a random matrix without the fourth moment condition

- Mathematics
- 2015

We obtain non-asymptotic lower bounds on the least singular value of ${\mathbf X}_{pn}^\top/\sqrt{n}$, where ${\mathbf X}_{pn}$ is a $p\times n$ random matrix whose columns are independent copies of… Expand

Delocalization of eigenvectors of random matrices. Lecture notes

- Mathematics
- 2017

Let $x \in S^{n-1}$ be a unit eigenvector of an $n \times n$ random matrix. This vector is delocalized if it is distributed roughly uniformly over the real or complex sphere. This intuitive notion… Expand

Upper bound for intermediate singular values of random matrices

- Mathematics
- 2016

In this paper, we prove that an $n\times n$ matrix $A$ with independent centered subgaussian entries satisfies \[ s_{n+1-l}(A) \le C_1t \frac{l}{\sqrt{n}} \] with probability at least… Expand