  Motivated by the future stability problem of solutions of the
Einstein--Yang--Mills (EYM) system with arbitrary dimension, we aim to (1)
construct a tensorial symmetric hyperbolic formulation for the
$(n+1)$-dimensional EYM system in the temporal gauge; (2) establish the local
well-posedness for the Cauchy problem of EYM equations in the temporal gauge
using this tensorial symmetric hyperbolic system. By introducing certain
auxiliary variables, we extend essentially the $(n+1)$-dimensional Yang--Mills
system to a tensorial symmetric hyperbolic system. On the contrary, this
symmetric hyperbolic system with data satisfying some constraints (extending
the Yang--Mills constraints) reduces to the Yang--Mills system. Consequently,
an equivalence between the EYM and the tensorial symmetric hyperbolic system
with a class of specific data set is concluded. Furthermore, a general
symmetric hyperbolic system over tensor bundles is studied, with which, we
conclude the local well-posedness of the EYM system. It turns out the idea of
symmetric hyperbolic formulation of the Yang--Mills field is very useful in
prompting a tensorial Fuchsian formalism and proving the future stability for
the EYM system with arbitrary dimension (i.e., this new symmetric hyperbolic
formulation of EYM manifests well behaved lower order terms for long time
evolution), see our companion article [21] with Todd A. Oliynyk.
