Strict localization of eigenvectors and eigenvalues

In this article we show and implement a simple and effcient method to strictly locate eigenvectors and eigenvalues of a given matrix, based on the modified cone condition. As a consequence we can also effectively localize zeros of complex polynomials.

💡 Research Summary

The paper introduces a novel and computationally efficient technique for rigorously localizing eigenvectors and eigenvalues of arbitrary matrices, based on a “modified cone condition.” Traditional eigenvalue bounding tools such as Gershgorin disks, Brauer ovals, or the Bauer‑Fike theorem often yield overly conservative regions, especially when eigenvalues are clustered. The authors address this limitation by defining ε‑cones—geometric cones in the vector space that are invariant or contractive under the action of the matrix. By decomposing a matrix A into its diagonal part D and off‑diagonal perturbation N, and assuming that the norm of N is sufficiently small relative to the diagonal entries, they prove three central results: (1) each ε‑cone associated with a diagonal entry d_i contains a unique eigenvalue λ_i, which lies in a narrow interval I_i around d_i; (2) if the intervals I_i are disjoint, the corresponding eigenvalues are guaranteed to be distinct; (3) the associated eigenvector v_i resides inside the same ε‑cone, providing a tight spatial bound on its direction.

The algorithm proceeds in four stages. First, the matrix is split into D and N, and a suitable matrix norm (∞‑norm or 2‑norm) of N is computed. Second, for each diagonal entry d_i, an admissible ε_i is derived as a function of ‖N‖ and the distance between |d_i| and ‖N‖, which determines the opening angle of the cone. Third, intervals I_i =