Eigenvalue Enclosures for Definite Hermitian Matrix Pencils under Structured Low-Rank Perturbations

Abstract

This study develops an eigenvalue-enclosure framework for definite Hermitian matrix pencils subject to simultaneous structured low-rank perturbations of both coefficient matrices. After reducing a definite pair to a positive-definite denominator, the perturbations are compressed to a common low-dimensional support. A shifted global enclosure is first obtained from Loewner inequalities. A projection-aware two-sided bound is then derived from the generalized Courant–Fischer principle; its index-dependent factors measure the overlap between the update support and lower or upper invariant subspaces. A rank-interlacing guard is intersected with these bounds. For selected eigenvalues, a determinant and inertia identity expresses the updated eigenvalue count through a correction matrix whose order equals the support dimension, enabling safeguarded bisection without a complete updated eigendecomposition. The construction is extended to general definite Hermitian pairs through a real rotation and a monotone Möbius map. In deterministic experiments on 216 real and complex Hermitian pencils (20,736 ordered eigenvalues), no coverage failure was observed and the combined intervals reduced the shifted global widths by 49.6% on average. A six-eigenvalue cluster was refined from mean initial width 2.23×10⁻¹ to 5.19×10⁻¹¹, and a localized mass–spring example produced a 73.1% mean reduction for selected modes. The method is most suitable for repeated low-rank updates when baseline spectral information is reusable and only selected eigenvalues are required.