On rigorous estimates of eigenspaces and eigenvalues of a matrix

TL;DR

This paper introduces a cone-based method inspired by hyperbolic dynamical systems for rigorously estimating eigenvalues and eigenspaces of matrices, including multiple and clustered eigenvalues, with improved accuracy over existing methods.

Contribution

The paper develops a novel cone-domination approach for precise spectral and eigenspace estimation, extending to multidimensional eigenspaces and eigenvalue clusters.

Findings

Provides rigorous bounds for eigenvalues and eigenspaces

Improves estimates for isolated eigenvalues

Handles multiple and clustered eigenvalues effectively

Abstract

We present a method of cones for rigorous estimations of eigenvectors, eigenspaces and eigenvalues of a matrix. The key notion is the cone-domination and is inspired by ideas from hyperbolic dynamical systems. We present theorems which allow to rigorously locate the spectrum of the matrix and the eigenspaces, also multidimensional ones in case of eigenvalues of multiplicity greater than one or clusters of close eigenvalues. In case of isolated eigenvalue we show that the our method give the same or better estimates than ones known in literature.