  We provide an elementary systematic discussion of single-trace matrix actions
and of the group of matrix reparameterization that acts on them. The action of
this group yields a generalized notion of gauge invariance which encompasses
ordinary diffeomorphism and gauge invariances. We apply the formalism to
non-abelian D-brane actions in arbitrary supergravity backgrounds, providing in
particular explicit checks of the consistency of Myers' formulas with
supergravity gauge invariances. We also draw interesting consequences for
emergent space models based on the study of matrix effective actions. For
example, in the case of the AdS_5XS^5 background, we explain how the standard
tensor transformation laws of the supergravity fields under ordinary
diffeomorphisms emerge from the D-instanton effective action in this
background.
