  The main purpose of this paper is providing a systematic study and
classification of non-scalar kernels for Reproducing Kernel Hilbert Spaces
(RKHS), to be used in the analysis of deformation in shape spaces endowed with
metrics induced by the action of groups of diffeomorphisms. After providing an
introduction to matrix-valued kernels and their relevant differential
properties, we explore extensively those, that we call TRI kernels, that induce
a metric on the corresponding Hilbert spaces of vector fields that is both
translation- and rotation-invariant. These are analyzed in an effective manner
in the Fourier domain, where the characterization of RKHS of curl-free and
divergence-free vector fields is particularly natural. A simple technique for
constructing generic matrix-valued kernels from scalar kernels is also
developed. We accompany the exposition of the theory with several examples, and
provide numerical results that show the dynamics induced by different choices
of TRI kernels on the manifold of labeled landmark points.
