相关论文: Normalitity preserving perturbations and augmentat…

A remarkable property of Hermitian ensembles is their universal behavior, that is, once properly rescaled the eigenvalue statistics does not depend on particularities of the ensemble. Recently, normal matrix ensembles have attracted…

Eigenvalues arising in scattering theory have been envisioned as a potential source of target signatures in nondestructive testing of materials, whereby perturbations of the eigenvalues computed for a penetrable medium would be used to…

Determining the effect of structural perturbations on the eigenvalue spectra of networks is an important problem because the spectra characterize not only their topological structures, but also their dynamical behavior, such as…

We analyze the consequences of different evolutions of the Hubble parameter on the spectrum of scalar inflationary perturbations. The analysis is restricted to inflationary phases described by a transient evolution, when uncommon features…

We consider higher-rank versions of the standard numerical range for matrices. A central motivation for this investigation comes from quantum error correction. We develop the basic structure theory for the higher-rank numerical ranges, and…

The Schur-Horn theorem is a well-known result that characterizes the relationship between the diagonal elements and eigenvalues of a symmetric (Hermitian) matrix. In this paper, we extend this theorem by exploring the eigenvalue…

Traditional numerical methods for calculating matrix eigenvalues are prohibitively expensive for high-dimensional problems. Iterative random sparsification methods allow for the estimation of a single dominant eigenvalue at reduced cost by…

Computing the eigenvectors and eigenvalues of a perturbed matrix can be remarkably difficult when the unperturbed matrix has repeated eigenvalues. In this work we show how the limiting eigenvectors and eigenvalues of a symmetric matrix…

We obtain general, exact formulas for the overlaps between the eigenvectors of large correlated random matrices, with additive or multiplicative noise. These results have potential applications in many different contexts, from quantum…

The characterization of the solution set for a class of algebraic Riccati inequalities is studied. This class arises in the passivity analysis of linear time invariant control systems. Eigenvalue perturbation theory for the Hamiltonian…

We study Hadamard variation of eigenvalues of Laplacian with respect to general domain perturbations. We show their existence up to the second order rigorously and characterize the derivatives, using associated eigenvalue problems in finite…

We solve the problem of determining the Weierstrass structure of a regular matrix pencil obtained by a low rank perturbation of another regular matrix pencil. We apply the result to find necessary and sufficient conditions for the existence…

In this paper, we explore the structure of the penultimate Gram matrix in deep neural networks, which contains the pairwise inner products of outputs corresponding to a batch of inputs. In several architectures it has been observed that…

Let $\mathbb{C}^{n\times n}$ be the set of all $n \times n$ complex matrices. For any Hermitian positive semi-definite matrices $A$ and $B$ in $\mathbb{C}^{n\times n}$, their new common upper bound less than $A+B-A:B$ is constructed, where…

For a quadratic matrix polynomial associated with a damped mass-spring system there are three types of critical eigenvalues, the eigenvalues $\infty$ and $0$ and the eigenvalues on the imaginary axis. All these are on the boundary of the…

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…

We prove existence results for optimization problems for the $k$th Laplace eigenvalue on closed Riemannian manifolds of dimension $m \geq 3$, depending on the choice of normalization. One such normalization leads to eigenvalue optimization…

Heavy-tailed fluctuations and power law statistics pervade physics, finance, and economics, yet their origin is often ascribed to systems poised near criticality. Here we show that such behavior can emerge far from instability through a…

For a given complex square matrix $A$ with constant row sum, we establish two new eigenvalue inclusion sets. Using these bounds, first we derive bounds for the second largest and smallest eigenvalues of adjacency matrices of $k$-regular…

This book is about solving matrix nearness problems that are related to eigenvalues or singular values or pseudospectra. These problems arise in great diversity in various fields, be they related to dynamics, as in questions of robust…