Related papers: Optimizing Gershgorin for Symmetric Matrices

Gershgorin's famous circle theorem states that all eigenvalues of a square matrix lie in disks (called Gershgorin disks) around the diagonal elements. Here we show that if the matrix entries are non-negative and an eigenvalue has geometric…

The application of the Gershgorin circle theorem and some of its derivatives to estimate the eigenvalues of a matrix is considered. The obtained results are developed to obtain the localization region of the eigenvalues of a matrix with…

A matrix is well separated if all its Gershgorin circles are away from the unit circle and they are separated from each other. In this article, the region of relative errors in the eigenvalues is obtained as a quadratic oval for non…

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 present Gerschgorin-type eigenvalue inclusion sets applicable to generalized eigenvalue problems.Our sets are defined by circles in the complex plane in the standard Euclidean metric, and are easier to compute than known similar…

Extending an earlier result for real matrices we show that multiple eigenvalues of a complex matrix lie in a reduced Gershgorin disk. One consequence is a slightly better estimate in the real case. Another one is a geometric application.…

Let ({\lambda}, v) be a known real eigenpair of a square real matrix A. In this paper it is shown how to locate the other eigenvalues of A in terms of the components of v. The obtained region is a union of Gershgorin discs of the second…

We consider the eigenvalue problem for the case where the input matrix is symmetric and its entries perturb in some given intervals. We present a characterization of some of the exact boundary points, which allows us to introduce an inner…

Due to their importance in both data analysis and numerical algorithms, low rank approximations have recently been widely studied. They enable the handling of very large matrices. Tight error bounds for the computationally efficient…

In this paper, we revisit the {\alpha}BB method for solving global optimization problems. We investigate optimality of the scaling vector used in Gerschgorin's inclusion theorem to calculate bounds on the eigenvalues of the Hessian matrix.…

In this paper, a new approach is presented to determine common eigenvalues of two matrices. It is based on Gerschgorin theorem and Bisection method. The proposed approach is simple and can be useful in image processing and noise estimation.

In this paper we shed more light on determinants of interval matrices. Computing the exact bounds on a determinant of an interval matrix is an NP-hard problem. Therefore, attention is first paid to approximations. NP-hardness of both…

We compare two established and a new method for the calculation of spectral bounds for Hessian matrices on hyperrectangles by applying them to a large collection of 1522 objective and constraint functions extracted from benchmark global…

Given a symmetric matrix $A$, we show from the simple sketch $GAG^T$, where $G$ is a Gaussian matrix with $k = O(1/\epsilon^2)$ rows, that there is a procedure for approximating all eigenvalues of $A$ simultaneously to within $\epsilon…

Matrix completion algorithms fill missing entries in a large matrix given a subset of observed samples. However, how to best pre-select informative matrix entries given a sampling budget is largely unaddressed. In this paper, we propose a…

We present a Gershgorin's type result on the localisation of the spectrum of a matrix. Our method is elementary and relies upon the method of Schur complements, furthermore it outperforms the one based on the Cassini ovals of Ostrovski and…

Let $T : \Omega \rightarrow \bbC^{n \times n}$ be a matrix-valued function that is analytic on some simply-connected domain $\Omega \subset \bbC$. A point $\lambda \in \Omega$ is an eigenvalue if the matrix $T(\lambda)$ is singular. In this…

Via the process of isospectral graph reduction the adjacency matrix of a graph can be reduced to a smaller matrix while its spectrum is preserved up to some known set. It is then possible to estimate the spectrum of the original matrix by…

This paper explores variants of the subspace iteration algorithm for computing approximate invariant subspaces. The standard subspace iteration approach is revisited and new variants that exploit gradient-type techniques combined with a…

We propose a fast general projection-free metric learning framework, where the minimization objective $\min_{\textbf{M} \in \mathcal{S}} Q(\textbf{M})$ is a convex differentiable function of the metric matrix $\textbf{M}$, and $\textbf{M}$…