Related papers: Asymptotically exact spectral estimates for left t…
We prove the Central Limit Theorem for finite-dimensional vectors of linear eigenvalue statistics of submatrices of Wigner random matrices under the assumption that test functions are sufficiently smooth. We connect the asymptotic…
We explore a certain family $\{A_n\}_{n=1}^{\infty}$ of $n \times n$ tridiagonal real symmetric matrices. After deriving a three-term recurrence relation for the characteristic polynomials of this family, we find a closed form solution. The…
In this paper we consider generalized eigenvalue problems for a family of operators with a quadratic dependence on a complex parameter. Our model is $L(\lambda)=-\triangle +(P(x)-\lambda)^2$ in $L^2(\R^d)$ where $P$ is a positive elliptic…
In this work, we study the spectral properties of matrix Hamiltonians generated by linearizing the nonlinear Schr\"odinger equation about soliton solutions. By a numerically assisted proof, we show that there are no embedded eigenvalues for…
We study properties of eigenvalues of a matrix associated with a randomly chosen partial automorphism of a regular rooted tree. We show that asymptotically, as the numbers of levels goes to infinity, the fraction of non-zero eigenvalues…
We consider spectra of $n$-by-$n$ irreducible tridiagonal matrices over a field and of their $n-1$-by-$n-1$ trailing principal submatrices. The real symmetric and complex Hermitian cases have been fully understood: it is necessary and…
We present a method to linearize, without approximation, a specific class of eigenvalue problems with eigenvector nonlinearities (NEPv), where the nonlinearities are expressed by scalar functions that are defined by a quotient of linear…
We obtain the asymptotic distribution of eigenvalues of real symmetric tridiagonal matrices as their dimension increases to infinity and whose diagonal and off-diagonal elements asymptotically change with the index n as J_{nt+i nt+i}\sim…
Formulas for matrix determinants, algebraic adjunctions, characteristic polynomial coefficients, components of eigenvectors are obtained in the form of signless sums of matrix elements products taking by special graphs. Signless formulas…
We describe an algorithm to compute the extremal eigenvalues and corresponding eigenvectors of a symmetric matrix by solving a sequence of Quadratic Binary Optimization problems. This algorithm is robust across many different classes of…
Many important problems are characterized by the eigenvalues of a large matrix. For example, the difficulty of many optimization problems, such as those arising from the fitting of large models in statistics and machine learning, can be…
We prove optimal convergence estimates for eigenvalues and eigenvectors of a class of singular/stiff perturbed problems. Our profs are constructive in nature and use (elementary) techniques which are of current interest in computational…
We analyze the correspondence between finite sequences of finitely supported probability distributions and finite-dimensional, real, symmetric, tridiagonal matrices. In particular, we give an intrinsic description of the topology induced on…
The statistics of eigenfunction amplitudes are studied in mesoscopic disordered electron systems of finite size. The exact eigenspectrum and eigenstates are obtained by solving numerically Anderson Hamiltonian on a three-dimensional lattice…
We propose a second-order accurate method to estimate the eigenvectors of extremely large matrices thereby addressing a problem of relevance to statisticians working in the analysis of very large datasets. More specifically, we show that…
We consider sample covariance matrices $S_N=\frac{1}{p}\Sigma_N^{1/2}X_NX_N^* \Sigma_N^{1/2}$ where $X_N$ is a $N \times p$ real or complex matrix with i.i.d. entries with finite $12^{\rm th}$ moment and $\Sigma_N$ is a $N \times N$…
A wide class of matrix pencils connected with dissipative Hamiltonian descriptor systems is investigated. In particular, the following properties are shown: all eigenvalues are in the closed left half plane, the nonzero finite eigenvalues…
We consider a class of linear eigenvalue problems depending on a small parameter epsilon in which the series expansion for the eigenvalue in powers of epsilon is divergent. We develop a new technique to determine the precise nature of this…
This paper is concerned with the interplay between statistical asymmetry and spectral methods. Suppose we are interested in estimating a rank-1 and symmetric matrix $\mathbf{M}^{\star}\in \mathbb{R}^{n\times n}$, yet only a randomly…
A Toeplitz matrix is one in which the matrix elements are constant along diagonals. The Fisher-Hartwig matrices are much-studied singular matrices in the Toeplitz family. The matrices are defined for all orders, $N$. They are parametrized…