<p>Dual matrices have emerged as a pivotal framework for modeling complex phenomena, such as neural traveling waves in brain science. Concurrently, the symmetric generalized eigenvalue problem remains a cornerstone of numerical linear algebra, particularly in high-dimensional techniques like linear discriminant analysis. This paper presents a comprehensive theoretical and algorithmic investigation of the generalized eigenvalue problem within the dual setting. We first elucidate the perturbation characteristics of dual Hermitian matrices, characterizing how their spectral properties change under small Hermitian perturbations. Building upon these insights, we develop algorithms for solving the dual symmetric generalized eigenvalue problem by leveraging the unitary decomposition of dual Hermitian matrices. To provide a more unified spectral framework, we further extend our analysis to the generalized eigenvalue problem for general dual complex matrices, establishing the necessary theoretical foundations for non-Hermitian cases. As a practical application of these advancements, we propose a Dual Linear Discriminant Analysis method designed to optimize class separation in dual-valued feature spaces. The efficacy and numerical reliability of our theoretical results and solvers are demonstrated through applications in hyperspectral remote sensing, facial recognition, and brain science.</p>