Deep matrix factorization (DMF) is a fundamental model underlying many applications, including deep linear neural networks. Despite its simplicity, the regularized DMF problem exhibits a highly nonconvex optimization landscape that is not yet fully understood. In this talk, we analyze the loss landscape and local geometry of regularized deep matrix factorization. We characterize all critical points and identify conditions a critical point is a local minimizer, a global minimizer, a strict saddle point, or a non-strict saddle point. We further establish an error bound around the critical point set, which leads to linear convergence guarantees for gradient-based methods. Our results provide theoretical insights into why first-order methods perform well for regularized DMF and offer a unified perspective on the optimization behavior of deep linear networks as an important application.