We develop projection pursuit for data that admit a natural representation in matrix form. For projection indices, we propose extensions of the classical kurtosis and Mardia’s multivariate kurtosis. The first index estimates projections for both sides of the matrices simultaneously, while the second index finds the two projections separately. Both indices are shown to recover the optimally separating projection for two-group Gaussian mixtures in the absence of any label information. We further establish the strong consistency of the corresponding sample estimators, as well as the asymptotic normality and high-dimensional consistency for the first estimator. Simulations and real data examples on hand-written postal codes and video data are used to demonstrate the method.