Related papers: The Kato--Temple inequality and eigenvalue concent…

On a finite connected metric graph, we establish upper bounds for the eigenvalues of the Laplacian. These bounds depend on the length, the Betti number, and the number of pendant vertices. For trees, these estimates are sharp. We also…

Classical matrix perturbation results, such as Weyl's theorem for eigenvalues and the Davis-Kahan theorem for eigenvectors, are general purpose. These classical bounds are tight in the worst case, but in many settings sub-optimal in the…

Standard perturbation theory of eigenvalue problems consists of obtaining approximations of eigenmodes in the neighborhood of an operator where the corresponding eigenmode is known. Nevertheless, if the corresponding eigenmodes of several…

What is the connection of random matrices with integrable systems? Is this connection really useful? Introducing apprpriate times in the distribution of the ensemble of matrices, one shows that the corresponding distribution of the…

Matrix perturbation bounds (such as Weyl and Davis-Kahan) are used abundantly in many areas of mathematics and data science. Many bounds (such as the above two) involve the spectral norm of the noise matrix and are sharp in worst case…

We study numerically and analytically the spectrum of incidence matrices of random labeled graphs on N vertices : any pair of vertices is connected by an edge with probability p. We give two algorithms to compute the moments of the…

We provide non-asymptotic, relative deviation bounds for the eigenvalues of empirical covariance and Gram matrices in general settings. Unlike typical uniform bounds, which may fail to capture the behavior of smaller eigenvalues, our…

We compute the limiting distributions of the largest eigenvalue of a complex Gaussian sample covariance matrix when both the number of samples and the number of variables in each sample become large. When all but finitely many, say $r$,…

We introduce a new random matrix model called distance covariance matrix in this paper, whose normalized trace is equivalent to the distance covariance. We first derive a deterministic limit for the eigenvalue distribution of the distance…

We propose a valid and consistent test for the hypothesis that two latent distance random graphs on the same vertex set have the same generating latent positions, up to some unidentifiable similarity transformations. Our test statistic is…

Random matrix theory allows one to deduce the eigenvalue spectrum of a large matrix given only statistical information about its elements. Such results provide insight into what factors contribute to the stability of complex dynamical…

An important facet of the inverse eigenvalue problem for graphs is to determine the minimum number of distinct eigenvalues of a particular graph. We resolve this question for the join of a connected graph with a path. We then focus on…

We consider eigenvectors of adjacency matrices of Erd\H{o}s-R\'{e}nyi graphs and study the variation of their directions by resampling the entries randomly. Let $\mathbf{v}$ be the eigenvector associated with the second-largest eigenvalue…

Quantifying the eigenvalue spectra of large random matrices allows one to understand the factors that contribute to the stability of dynamical systems with many interacting components. This work explores the effect that the interaction…

Using Random Matrix Theory one can derive exact relations between the eigenvalue spectrum of the covariance matrix and the eigenvalue spectrum of its estimator (experimentally measured correlation matrix). These relations will be used to…

Eigenvalue interlacing is a versatile technique for deriving results in algebraic combinatorics. In particular, it has been successfully used for proving a number of results about the relation between the (adjacency matrix or Laplacian)…

How many samples are sufficient to guarantee that the eigenvectors and eigenvalues of the sample covariance matrix are close to those of the actual covariance matrix? For a wide family of distributions, including distributions with finite…

This work deals with the isogeometric Galerkin discretization of the eigenvalue problem related to the Laplace operator subject to homogeneous Dirichlet boundary conditions on bounded intervals. This paper uses GLT theory to study the…

We apply eigenvalue interlacing techniques for obtaining lower and upper bounds for the sums of Laplacian eigenvalues of graphs, and characterize equality. This leads to generalizations of, and variations on theorems by Grone, and Grone and…

We confirm universal behaviors such as eigenvalue distribution and spacings predicted by Random Matrix Theory (RMT) for the cross correlation matrix of the daily stock prices of Tokyo Stock Exchange from 1993 to 2001, which have been…