The present article is concerned with the interaction of solitary wave solutions of the matrix Korteweg-de Vries equation. The picture is essentially richer than in the classical scalar case since collisions may be less elasticin the sense that they do not only cause a position shift but also a change of shape. Our construction of solutions is based on a general solution formula with matrix parameters. After a discussion which parameters yield solutions con-sisting of one localized wave, an asymptotic description is obtained for the interaction of two such waves, the crucial point being explicit formulas for the change of shape. The main result extends previous work by Goncharenko, who studied waves coming from matrices of rank 1. As usual, it is to be expected that nonlinear superpositions of finitely many waves behave like combinations of 2-soliton interactions.