Related papers: The low-rank eigenvalue problem

Eigenvector localization refers to the situation when most of the components of an eigenvector are zero or near-zero. This phenomenon has been observed on eigenvectors associated with extremal eigenvalues, and in many of those cases it can…

The standard approach for finding eigenvalues and eigenvectors of matrix polynomials starts by embedding the coefficients of the polynomial into a matrix pencil, known as linearization. Building on the pioneering work of Nakatsukasa and…

In this paper we study the structure and give bounds for the eigenvalues of the $n\times n$ matrix, which $ij$ entry is $(i,j)^\alpha[i,j]^\beta$, where $\alpha,\beta\in\Rset$, $(i,j)$ is the greatest common divisor of $i$ and $j$ and…

This paper describes a set of rational filtering algorithms to compute a few eigenvalues (and associated eigenvectors) of non-Hermitian matrix pencils. Our interest lies in computing eigenvalues located inside a given disk, and the proposed…

Following the Perron-Frobenius theorem, the spectral radius of a primitive matrix is a simple eigenvalue. It is shown that for a primitive matrix $A$, there is a positive rank one matrix $X$ such that $B = A \circ X$, where $\circ$ denotes…

In the current work, we study the eigenvalue distribution results of a class of non-normal matrix-sequences which may be viewed as a low rank perturbation, depending on a parameter $\beta>1$, of the basic Toeplitz matrix-sequence…

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.…

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…

The properties of the first (largest) eigenvalue and its eigenvector (first eigenvector) are investigated for large sparse random symmetric matrices that are characterized by bimodal degree distributions. In principle, one should be able to…

In the last decade matrix polynomials have been investigated with the primary focus on adequate linearizations and good scaling techniques for computing their eigenvalues and eigenvectors. In this article we propose a new method for…

Let $m,n>1$ be integers and $\mathbb{P}_{n,m}$ be the point set of the projective $(n-1)$-space (defined by [2]) over the ring $\mathbb{Z}_m$of integers modulo $m$. Let $A_{n,m}=(a_{uv})$ be the matrix with rows and columns being labeled by…

A unified theory of orthogonal polynomials of a discrete variable is presented through the eigenvalue problem of hermitian matrices of finite or infinite dimensions. It can be considered as a matrix version of exactly solvable Schr\"odinger…

We study the eigenvalue problem for some special class of anti-triangular matrices. Though the eigenvalue problem is quite classical, as far as we know, almost nothing is known about properties of eigenvalues for anti-triangular matrices.…

Riordan matrices are infinite lower triangular matrices determined by a pair of formal power series over the real or complex field. These matrices have been mainly studied as combinatorial objects with an emphasis placed on the algebraic or…

We consider an iterative eigensolver for Schr\"odinger equations that constructs low-rank approximations of eigenfunctions with accuracy-adapted ranks, with particular focus on fermionic Schr\"odinger equations in second-quantized form and…

Low-rank approximation with zeros aims to find a matrix of fixed rank and with a fixed zero pattern that minimizes the Euclidean distance to a given data matrix. We study the critical points of this optimization problem using algebraic…

The study of solving the inverse eigenvalue problem for nonnegative matrices has been around for decades. It is clear that an inverse eigenvalue problem is trivial if the desirable matrix is not restricted to a certain structure. Provided…

In this paper, we present a determinist Jordan normal form algorithms based on the Fadeev formula: \[(\lambda \cdot I-A) \cdot B(\lambda)=P(\lambda) \cdot I\] where $B(\lambda)$ is $(\lambda \cdot I-A)$'s comatrix and $P(\lambda)$ is $A$'s…

We characterize the eigenvalues and eigenvectors of a class of complex valued tridiagonal $n$ by $n$ matrices subject to arbitrary boundary conditions, i.e. with arbitrary elements on the first and last rows of the matrix. %By boundary…

Nonlinear eigenvalue problems with eigenvector nonlinearities (NEPv) are algebraic eigenvalue problems whose matrix depends on the eigenvector. Applications range from computational quantum mechanics to machine learning. Due to its…