Related papers: Eigenvalue analysis of constrained minimization pr…
Horn's problem, i.e., the study of the eigenvalues of the sum $C=A+B$ of two matrices, given the spectrum of $A$ and of $B$, is re-examined, comparing the case of real symmetric, complex Hermitian and self-dual quaternionic $3\times 3$…
In this article in a very general manner we have investigated the eigen value problem in Rindler space. We have developed the formalism in an exact form. It has been noticed that although the Hamiltonian is non-hermitian, because of the…
We give an introduction to the theory and to some applications of eigenvectors of tensors (in other words, invariant one-dimensional subspaces of homogeneous polynomial maps), including a review of some concepts that are useful for their…
Let $\mathcal{A}(H)$ be the adjacency tensor of $r$-uniform hypergraph $H$. If $H$ is connected, the unique positive eigenvector $x=(x_1,x_2,\ldots,x_n)^{\mathrm{T}}$ with $||x||_r=1$ corresponding to spectral radius $\rho(H)$ is called the…
A technique for constructing an infinite tower of pairs of PT-symmetric Hamiltonians, $\hat{H}_n$ and $\hat{K}_n$ (n=2,3,4,...), that have exactly the same eigenvalues is described. The eigenvalue problem for the first Hamiltonian…
We prove that multilinear (tensor) analogues of many efficiently computable problems in numerical linear algebra are NP-hard. Our list here includes: determining the feasibility of a system of bilinear equations, deciding whether a 3-tensor…
The purpose of this paper is to prove that the spectrum of the non-self-adjoint one-particle Hamiltonian proposed by J. Feinberg and A. Zee (Phys. Rev. E 59 (1999), 6433--6443) has interior points. We do this by first recalling that the…
For a Hermitian matrix $H \in \mathbb C^{n,n}$ and symmetric matrices $S_0, S_1,\ldots,S_k \in \mathbb C^{n,n}$, we consider the problem of computing the supremum of $\left\{ \frac{v^*Hv}{v^*v}:~v\in \mathbb C^{n}\setminus…
For a nonnegative symmetric weakly irreducible tensor, its spectral radius is an eigenvalue corresponding to a unique positive eigenvector up to a scalar called the Perron vector. But including the Perron vector, there may have more than…
This article introduces an algebraic framework for establishing eigenvalue bounds for symmetric positive definite tensors by leveraging intrinsic invariants, specifically the trace and determinant (resultant). We derive a hierarchy of…
We study the problem of determining whether a prescribed eigenpair $(\lambda,x)$ can be made an exact eigenpair of a nonnegative Hankel matrix through the smallest possible structured perturbation. The task reduces to check the feasibility…
A new measure called min-max elementwise backward error is introduced for approximate roots of scalar polynomials $p(z)$. Compared with the elementwise relative backward error, this new measure allows for larger relative perturbations on…
A polynomial matrix inequality is a formula asserting that a polynomial matrix is positive semidefinite. Polynomial matrix optimization concerns minimizing the smallest eigenvalue of a symmetric polynomial matrix subject to a tuple of…
In this paper, we discuss approximating the eigenvalue problem of biharmonic equation. We first present an equivalent mixed formulation which admits amiable nested discretization. Then, we construct multi-level finite element schemes by…
We consider the problem of finding nonzero eigenvalues and the corresponding eigenvectors of a matrix $AA^{\top}$, where $A$ is a special incidence matrix; This matrix can equivalently be defined based on a match relation between some…
One of the most used approaches in simulating materials is the tight-binding approximation. When using this method in a material simulation, it is necessary to compute the eigenvalues and eigenvectors of the Hamiltonian describing the…
A recent conjecture regarding the average of the minimum eigenvalue of the reduced density matrix of a random complex state is proved. In fact, the full distribution of the minimum eigenvalue is derived exactly for both the cases of a…
We consider problems related to the asymptotic minimization of eigenvalues of anisotropic harmonic oscillators in the plane. In particular we study Riesz means of the eigenvalues and the trace of the corresponding heat kernels. The…
Symmetric tensor decomposition is an important problem with applications in several areas for example signal processing, statistics, data analysis and computational neuroscience. It is equivalent to Waring's problem for homogeneous…
In this paper a novel numerical approximation of parametric eigenvalue problems is presented. We motivate our study with the analysis of a POD reduced order model for a simple one dimensional example. In particular, we introduce a new…