Related papers: Gaussian Regularization of the Pseudospectrum and …

Consider the matrix $A_{\mathcal{G}}$ chosen uniformly at random from the finite set of all $N$-dimensional matrices of zero main-diagonal and binary entries, having each row and column of $A_{\mathcal{G}}$ sum to $d$. That is, the…

The compression of a matrix $A\in\mathbb C^{n\times n}$ onto a subspace $V\subset\mathbb C^n$ is the matrix $Q^*AQ$ where the columns of $Q$ form an orthonormal basis for $V$. This is an important object in both operator theory and…

We study the angles between the eigenvectors of a random $n\times n$ complex matrix $M$ with density $\propto \mathrm{e}^{-n\operatorname{Tr}V(M^*M)}$ and $x\mapsto V(x^2)$ convex. We prove that for unit eigenvectors…

Random Hermitian matrices with a source term arise, for instance, in the study of non-intersecting Brownian walkers \cite{Adler:2009a, Daems:2007} and sample covariance matrices \cite{Baik:2005}. We consider the case when the $n\times n$…

Much effort has been spent on characterizing the spectrum of the non-backtracking matrix of certain classes of graphs, with special emphasis on the leading eigenvalue or the second eigenvector. Much less attention has been paid to the…

Given a symmetric matrix $M$ and a vector $\lambda$, we present new bounds on the Frobenius-distance utility of the Gaussian mechanism for approximating $M$ by a matrix whose spectrum is $\lambda$, under $(\varepsilon,\delta)$-differential…

For a finite simple undirected graph $G$, let $\gamma(G)$ denote the size of a smallest dominating set of $G$ and $\mu(G)$ denote the number of eigenvalues of the Laplacian matrix of $G$ in the interval $[0,1)$, counting multiplicities.…

Suppose $G$ is a controllable graph of order $n$ with adjacency matrix $A$. Let $W=[e,Ae,\ldots,A^{n-1}e]$ ($e$ is the all-one vector) and $\Delta=\prod_{i>j}(\alpha_i-\alpha_j)^2$ ($\alpha_i$'s are eigenvalues of $A$) be the walk matrix…

Let $A_1, \ldots, A_r$ be Hermitian $n \times n$ matrices and $S = \sum A_i \otimes s_i$ the associated matrix semicircle, where $s_1, \ldots, s_r$ are free semicircular variables. We prove that the following are equivalent: (i) the matrix…

Given a real-valued positive semidefinite matrix, Williamson proved that it can be diagonalised using symplectic matrices. The corresponding diagonal values are known as the symplectic spectrum. This paper is concerned with the stability of…

For a nonsingular integer matrix A, we study the growth of the order of A modulo N. We say that a matrix is exceptional if it is diagonalizable, and a power of the matrix has all eigenvalues equal to powers of a single rational integer, or…

Rayleigh-Schr\"{o}dinger perturbation theory is a well-known theory in quantum mechanics and it offers useful characterization of eigenvectors of a perturbed matrix. Suppose $A$ and perturbation $E$ are both Hermitian matrices, $A^t = A +…

Given a symmetric matrix $A$, we show from the simple sketch $GAG^T$, where $G$ is a Gaussian matrix with $k = O(1/\epsilon^2)$ rows, that there is a procedure for approximating all eigenvalues of $A$ simultaneously to within $\epsilon…

We show that for an $n\times n$ random symmetric matrix $A_n$, whose entries on and above the diagonal are independent copies of a sub-Gaussian random variable $\xi$ with mean $0$ and variance $1$, \[\mathbb{P}[s_n(A_n) \le…

We revisit the operator mixing in massless QCD-like theories. In particular, we address the problem of determining under which conditions a renormalization scheme exists where the renormalized mixing matrix in the coordinate representation,…

For any $\alpha\in (0,1)$ and any $n^{\alpha}\leq d\leq n/2$, we show that $\lambda(G)\leq C_\alpha \sqrt{d}$ with probability at least $1-\frac{1}{n}$, where $G$ is the uniform random $d$-regular graph on $n$ vertices, $\lambda(G)$ denotes…

It is shown that certain ensembles of random matrices with entries that vanish outside a band around the diagonal satisfy a localization condition on the resolvent which guarantees that eigenvectors have strong overlap with a vanishing…

We consider a class of sparse random matrices of the form $A_n =(\xi_{i,j}\delta_{i,j})_{i,j=1}^n$, where $\{\xi_{i,j}\}$ are i.i.d.~centered random variables, and $\{\delta_{i,j}\}$ are i.i.d.~Bernoulli random variables taking value $1$…

Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of the degrees of $G$. For any real $\alpha\in [0,1]$, Nikiforov \cite{VN1} defined the matrix $A_{\alpha}(G)$ as $$A_{\alpha}(G)=\alpha…

We develop a theoretical approach to compute the conditioned spectral density of $N \times N$ non-invariant random matrices in the limit $N \rightarrow \infty$. This large deviation observable, defined as the eigenvalue distribution…