Investigating the dynamics of learning in machine learning algorithms is of paramount importance for understanding how and why an approach may be successful. The tools of physics and statistics provide a robust setting for such investigations. Here, we apply concepts from random matrix theory to describe stochastic weight matrix dynamics, using the framework of Dyson Brownian motion. We derive the linear scaling rule between the learning rate (step size) and the batch size, and identify universal and non-universal aspects of weight matrix dynamics. We test our findings in the (near-)solvable case of the Gaussian Restricted Boltzmann Machine and in a linear one-hidden-layer neural network.