Istkhar Ali, Mohammad Waquas Khadim2025-11-172025978-93-6884-936-0http://136.232.12.194:4000/handle/123456789/1555In Advanced Studies in Mathematics and Statistics Editor: Mobin Ahmad"This chapter presents a comprehensive exploration of perturbation theory and eigenvalue localization for matrices over the quaternion division algebra, addressing key theoretical and computational challenges arising from their non-commutative nature. Building on recent advances in quaternionic linear algebra, we extend classical perturbation results-such as the Bauer-Fike theorem and relative eigenvalue bounds-to the quaternionic setting, with a focus on both right and left eigenvalues, which exhibit fundamentally distinct behaviors. We derive new perturbation bounds for structured quaternionic matrices (e.g., Hermitian, unitary, and skew-Hermitian), leveraging their unique spectral properties."en-USMATHEMATICSBook chapter