Related papers: Eigenvalue placement for regular matrix pencils wi…

Let A(z) be an analytic square matrix and $\lambda_{0}$ an eigenvalue of A(0) of multiplicity m. Then under the generic condition, the characteristic polynomial of A(z) evaluated at $\lambda_{0}$ has a simple zero at z=0, we prove that the…

Consider a real diagonal deterministic matrix $X_n$ of size $n$ with spectral measure converging to a compactly supported probability measure. We perturb this matrix by adding a random finite rank matrix, with delocalized eigenvectors. We…

We characterize possible spectra of rank-one perturbations B of a self-adjoint operator A with discrete spectrum and, in particular, prove that the spectrum of B may include any number of real or non-real eigenvalues of arbitrary algebraic…

We consider a non-compact Riemannian periodic manifold such that the corresponding Laplacian has a spectral gap. By continuously perturbing the periodic metric locally we can prove the existence of eigenvalues in a gap. A lower bound on the…

The inverse spectral problem for the second-order differential pencil with quadratic dependence on the spectral parameter is studied. We obtain sufficient conditions for the global solvability of the inverse problem, prove its local…

We compute the limiting eigenvalue statistics at the edge of the spectrum of large Hermitian random matrices perturbed by the addition of small rank deterministic matrices. To be more precise, we consider random Hermitian matrices with…

Our goal is to study statistical properies of "dielectric resonances" which are poles of conductance of a large random $LC$ network. Such poles are a particular example of eigenvalues $\lambda_n$ of matrix pencils ${\bf H}-\lambda {\bf W}$,…

This paper presents a fast, randomized divide-and-conquer algorithm for the definite generalized eigenvalue problem, which corresponds to pencils $(A,B)$ in which $A$ and $B$ are Hermitian and the Crawford number $\gamma(A,B) =…

In [M. Walter et al., Science 340, 1205, 7 June (2013)], they gave a sufficient condition for genuinely entangled pure states and discussed SLOCC classification via polytopes and the eigenvalues of the single-particle states. In this paper,…

In this text, based on elementary computations, we provide a perturbative expansion of the coordinates of the eigenvectors of a Hermitian matrix of large size perturbed by a random matrix with small operator norm whose entries in the…

A large i.i.d. random matrix with deterministic low-rank perturbation has been extensively studied, particularly in the aspects of the ESD (Empirical Spectral Distribution) and the outliers of eigenvalues. In this work, we investigate the…

Block full rank pencils introduced in [Dopico et al., Local linearizations of rational matrices with application to rational approximations of nonlinear eigenvalue problems, Linear Algebra Appl., 2020] allow us to obtain local information…

Matrix pencils, or pairs of matrices, may be used in a variety of applications. In particular, a pair of matrices (E,A) may be interpreted as the differential equation E x' + A x = 0. Such an equation is invariant by changes of variables,…

A number of theoretical and computational problems for matrix polynomials are solved by passing to linearizations. Therefore a perturbation theory results for linearizations need to be related back to matrix polynomials. In this paper we…

We consider the eigenvalues and eigenvectors of finite, low rank perturbations of random matrices. Specifically, we prove almost sure convergence of the extreme eigenvalues and appropriate projections of the corresponding eigenvectors of…

In this work, we consider symmetric positive definite pencils depending on two parameters. That is, we are concerned with the generalized eigenvalue problem $A(x)-\lambda B(x)$, where $A$ and $B$ are symmetric matrix valued functions in…

We establish a large deviation principle for the largest eigenvalue of a rank one deformation of a matrix from the GUE or GOE. As a corollary, we get another proof of the phenomenon, well-known in learning theory and finance, that the…

We present first-order perturbation analysis of a simple eigenvalue and the corresponding right and left eigenvectors of a general square matrix, not assumed to be Hermitian or normal. The eigenvalue result is well known to a broad…

For piecewise expanding one-dimensional maps without periodic turning points we prove that isolated eigenvalues of small (random) perturbations of these maps are close to isolated eigenvalues of the unperturbed system. (Here ``eigenvalue''…

In this paper we show how to construct diagonal scalings for arbitrary matrix pencils $\lambda B-A$, in which both $A$ and $B$ are complex matrices (square or nonsquare). The goal of such diagonal scalings is to "balance" in some sense the…