  In this paper, we discuss the solution of certain matrix-valued partial
differential equations. Such PDEs arise, for example, when constructing a
Riemannian contraction metric for a dynamical system given by an autonomous
ODE. We develop and analyse a new meshfree discretisation scheme using
kernel-based approximation spaces. However, since these approximation spaces
have now to be matrix-valued, the kernels we need to use are fourth order
tensors. We will review and extend recent results on even more general
reproducing kernel Hilbert spaces. We will then apply this general theory to
solve a matrix-valued PDE and derive error estimates for the approximate
solution. The paper ends with a typical example from dynamical systems.
