Matrix perturbation theory is a foundational subject across multiple disciplines, including probability, statistics, machine learning, and applied mathematics. Perturbation bounds play a critical role in quantifying the impact of small noise on matrix spectral parameters in various applications such as matrix completion, principal component analysis, and community detection. In this talk, we focus on the additive perturbation model, where a low-rank data matrix is perturbed by Gaussian noise. We provide a comprehensive analysis of the singular subspaces, extending the classical Davis-Kahan-Wedin theorem and offering fine-grained analysis of the singular vector matrices. Also, we will show the practical implications of our perturbation results, specifically highlighting their application in Gaussian mixture models. The talk is based on joint works with Sean O'Rourke and Van Vu.