The Primal-Dual Method of Multipliers (PDMM) is a new algorithm that solves convex optimization problems in a distributed manner. This study focuses on the convergence behavior of the PDMM. For a deeper understanding, the PDMM algorithm was applied to distributed averaging and distributed dictionary learning problems. The results were compared to those of other state-of-the-art algorithms. The experiments show that the PDMM algorithm not only has a fast convergence rate but also robust performance against transmission failures in the network. Furthermore, on the basis of these experiments, the convergence rate of the PDMM was analyzed. Different attempts at proving the linear convergence rate were carried out. As a result, the linear convergence rate has been proven under certain conditions.