Recently, Sharpness-Aware Minimization (SAM) has gained widespread attention for effectively improving generalization performance. We begin by establishing a theoretical connection between the top eigenvalue of the Hessian matrix and generalization error using an extended PAC-Bayes theorem. Building on this foundation, we derive a third-order stochastic differential equation (SDE) to model the dynamics of SAM, which reveals a lower approximation error compared to previous second-order SDE approaches. Our theoretical analysis highlights the significance of perturbation-eigenvector alignment in reducing sharpness. To address the practical challenges of achieving this alignment, we introduce Eigen-SAM, which intermittently estimates the top eigenvector and enhances alignment, resulting in better sharpness minimization. We validate our theoretical insights and the effectiveness of Eigen-SAM through extensive experiments across multiple datasets and model architectures, demonstrating consistent improvements in test accuracy and robustness over standard SAM and SGD.