The Generalized Eigenvalue Proximal Support Vector Machine (GEPSVM) introduces a novel large-margin classifier that improves upon standard SVMs by constructing a pair of non-parallel hyperplanes derived from a generalized eigenvalue problem. However, the GEPSVM suffers from severe misclassification in the overlapped hyperplane region, known as the underdetermined hyperplane problem (UHP). A localized GEPSVM (LGEPSVM) alleviates this issue by building convex hulls on the hyperplanes for classification, but it still faces notable drawbacks: (1) an inability to integrate both local and global information, (2) a lack of consideration of the data’s statistical characteristics, and (3) high computational and storage costs. To address these limitations, we propose the Ellipsoid-structured Localized GEPSVM (EL-GEPSVM), which extends the GEPSVM by constructing ellipsoid-structured convex hulls under the Mahalanobis metric. This design incorporates statistical data characteristics and enables a classification scheme that simultaneously considers local and global information. Extensive theoretical analyses and experiments demonstrate that the proposed EL-GEPSVM achieves improved effectiveness and efficiency compared with existing methods.