Related papers: The Quantum Eigenvalue Problem and Z-Eigenvalues o…

Tensor eigenvalues and eigenvectors have been introduced in the recent mathematical literature as a generalization of the usual matrix eigenvalues and eigenvectors. We apply this formalism to a tensor that describes a multipartite symmetric…

The purpose of this paper is to study the problem of computing unitary eigenvalues (U-eigenvalues) of non-symmetric complex tensors. By means of symmetric embedding of complex tensors, the relationship between U-eigenpairs of a…

In this paper, the concepts of Pareto $H$-eigenvalue and Pareto $Z$-eigenvalue are introduced for studying constrained minimization problem and the necessary and sufficient conditions of such eigenvalues are given. It is proved that a…

This paper addresses two fundamental problems posed by Qi regarding the sufficiency of eigenvalues for the classification of symmetric tensors in the two-dimensional setting. For $2\times2\times2$ and $2\times2\times2\times2$ complex…

Real eigenpairs of a real antisymmetric tensor of order $p$ and dimension $N$ can be defined as pairs of a real eigenvalue and $p$ orthonormal $N$-dimensional real eigenvectors. We compute the signed and the genuine distributions of such…

For integers $m\geq 3$ and $1\leq\ell\leq m-1$, we study the eigenvalue problems $-u^{\prime\prime}(z)+[(-1)^{\ell}(iz)^m-P(iz)]u(z)=\lambda u(z)$ with the boundary conditions that $u(z)$ decays to zero as $z$ tends to infinity along the…

A real number $\lambda$ is called a Z-eigenvalue of a tensor $A$, if $\lambda$ is an eigenvalue of $A$ and the corresponding eigenvector $v$ is real and satisfies $v^Tv=1$. In this paper we compute the expected number of Z-eigenvalues of a…

An $n \times n \times p$ tensor is called a T-square tensor. It arises from many applications, such as the image feature extraction problem and the multi-view clustering problem. We may symmetrize a T-square tensor to a T-symmetric tensor.…

The probability that there are $k$ real eigenvalues for an $n$ dimensional real random matrix is known. Here we study this for the case of products of independent random matrices. Relating the problem of the probability that the product of…

This paper discusses the computation of real Z-eigenvalues and H-eigenvalues of nonsymmetric tensors. A general nonsymmetric tensor has finitely many Z-eigenvalues, while there may be infinitely many ones for special tensors. In the…

Eigenvectors of tensors, as studied recently in numerical multilinear algebra, correspond to fixed points of self-maps of a projective space. We determine the number of eigenvectors and eigenvalues of a generic tensor, and we show that the…

Quantum physics is generally concerned with real eigenvalues due to the unitarity of time evolution. With the introduction of $\mathcal{PT}$ symmetry, a widely accepted consensus is that, even if the Hamiltonian of the system is not…

This paper studies how to compute all real eigenvalues of a symmetric tensor. As is well known, the largest or smallest eigenvalue can be found by solving a polynomial optimization problem, while the other middle eigenvalues can not. We…

Nonlinear eigenvalue equations arise naturally in quantum information theory, particularly in the variational quantification of entanglement. In this work, we present a hybrid analytical and numerical framework for evaluating the geometric…

We study the eigenvalue problem -u"(z)-[(iz)^m+P(iz)]u(z)=\lambda u(z) with the boundary conditions that u(z) decays to zero as z tends to infinity along the rays \arg z=-\frac{\pi}{2}\pm \frac{2\pi}{m+2}, where P(z)=a_1 z^{m-1}+a_2…

The geometric measure of entanglement is a widely used entanglement measure for quantum pure states. The key problem of computation of the geometric measure is to calculate the entanglement eigenvalue, which is equivalent to computing the…

We study the eigenvalue problem $-u^{\prime\prime}(z)-[(iz)^m+P_{m-1}(iz)]u(z)=\lambda u(z)$ with the boundary conditions that $u(z)$ decays to zero as $z$ tends to infinity along the rays $\arg z=-\frac{\pi}{2}\pm \frac{2\pi}{m+2}$, where…

Let $n$ be a positive integer and $m$ be a positive even integer. Let ${\mathcal A}$ be an $m^{th}$ order $n$-dimensional real weakly symmetric tensor and ${\mathcal B}$ be a real weakly symmetric positive definite tensor of the same size.…

Einstein's field equations of gravitation are known to admit closed timelike curve (CTC) solutions. Deutsch approached the problem from the quantum information point of view and proposed a self-consistency condition. In this work, the…

An analytical perturbative method is suggested for solving the Helmholtz equation (\bigtriangledown^{2} + k^{2}){\psi} = 0 in two dimensions where {\psi} vanishes on an irregular closed curve. We can thus find the energy levels of a quantum…