Related papers: Delocalization of eigenvectors of random matrices.…

We consider the standard overlap $\mathcal{O}_{ij}: =\langle \mathbf{r}_j, \mathbf{r}_i\rangle\langle \mathbf{l}_j, \mathbf{l}_i\rangle$ of any bi-orthogonal family of left and right eigenvectors of a large random matrix $X$ with centred…

Let $X_N$ be a $N\times N$ matrix whose entries are i.i.d. complex random variables with mean zero and variance $\frac{1}{N}$. We study the asymptotic spectral distribution of the eigenvalues of the covariance matrix $X_N^*X_N$ for…

Using our previously published algorithm, we analyze the eigenvectors of the generalized Laplacian for two metric graphs occurring in practical applications. As expected, localization of an eigenvector is rare and the network should be…

In this paper we consider $N \times N $ matrices $D_{N}$ with i.i.d. entries all following an $a-$stable law divided by $N^{1/a}$. We prove that the least singular value of $D_{N}$, multiplied by $N$, tends to the same law as in the…

We consider $N\times N$ symmetric random matrices where the probability distribution for each matrix element is given by a measure $\nu$ with a subexponential decay. We prove that the eigenvalue spacing statistics in the bulk of the…

We study random, symmetric $N \times N$ band matrices with a band of size $W$ and Bernoulli random variables as entries. This interpolates between nearest neighbour interaction $W = 1$ and Wigner matrices $W = N$. Eigenvectors are known to…

We consider inhomogeneous square random matrices of size $N$ with independent entries of mean 0 and finite variance. We assume that the variance profile of this matrix is doubly stochastic and has a band-like structure with an appropriately…

Suppose that there is a family of $n$ random points $X_v$ for $v \in V$, independently and uniformly distributed in the square $\left[-\sqrt{n}/2,\sqrt{n}/2\right]^2$ of area $n$. We do not see these points, but learn about them in one of…

We consider random matrices of the form $H = W + \lambda V$, $\lambda\in\mathbb{R}^+$, where $W$ is a real symmetric or complex Hermitian Wigner matrix of size $N$ and $V$ is a real bounded diagonal random matrix of size $N$ with i.i.d.\…

We generalise the inference procedure for eigenvectors of symmetrizable matrices of Tyler (1981) to that of invariant and singular subspaces of non-diagonalizable matrices. Wald tests for invariant vectors and $t$-tests for their individual…

The Eigendecomposition of quadratic forms (symmetric matrices) guaranteed by the spectral theorem is a foundational result in applied mathematics. Motivated by a shared structure found in inferential problems of recent interest---namely…

In this work we present a framework for studying the eigenvalues of a family of matrices with a particular displacement structure. The family admits a specific decomposition as the product of an upper and a lower triangular matrices having…

The spectral properties of the adjacency matrix, in particular its largest eigenvalue and the associated principal eigenvector, dominate many structural and dynamical properties of complex networks. Here we focus on the localization…

We develop a theory for the eigenvalue density of arbitrary non-Hermitian Euclidean matrices. Closed equations for the resolvent and the eigenvector correlator are derived. The theory is applied to the random Green's matrix relevant to wave…

We study the spectral measure of large Euclidean random matrices. The entries of these matrices are determined by the relative position of $n$ random points in a compact set $\Omega_n$ of $\R^d$. Under various assumptions we establish the…

A number of inference problems with sensor networks involve projecting a measured signal onto a given subspace. In existing decentralized approaches, sensors communicate with their local neighbors to obtain a sequence of iterates that…

Let $T : \Omega \rightarrow \bbC^{n \times n}$ be a matrix-valued function that is analytic on some simply-connected domain $\Omega \subset \bbC$. A point $\lambda \in \Omega$ is an eigenvalue if the matrix $T(\lambda)$ is singular. In this…

We consider $n\times n$ real symmetric and hermitian random matrices $H_{n,m}$ equals the sum of a non-random matrix $H_{n}^{(0)}$ matrix and the sum of $m$ rank-one matrices determined by $m$ i.i.d. isotropic random vectors with…

We consider a square random matrix of size N of the form A + Y where A is deterministic and Y has iid entries with variance 1/N. Under mild assumptions, as N grows, the empirical distribution of the eigenvalues of A+Y converges weakly to a…

We study eigenvectors in the deformed Gaussian unitary ensemble of random matrices $H=W\tilde{H}W$, where $\tilde{H}$ is a random matrix from Gaussian unitary ensemble and $W$ is a deterministic diagonal matrix with positive entries. Using…