  We first investigate the structure of the systems derived from the gPC based
stochastic Galerkin method for the nonlinear hyperbolic systems with random
inputs. This method adopts a generalized Polynomial Chaos (gPC) approximations
in the stochastic Galerkin framework, but such approximations to the nonlinear
hyperbolic systems do not necessarily yield hyperbolic systems
\cite{Lucor2013}. Thus based on the work in \cite{framework}, we propose a
framework to carry out the model reduction for the general nonlinear hyperbolic
system to derive a final global system. Within this framework, the nonlinear
hyperbolic system in one space dimension and the symmetric hyperbolic system in
multiple space dimensions are reduced into a symmetric hyperbolic system based
on the stochastic Galerkin method. We note that the basis functions in the
expansion are not restricted to the random-dependent polynomials as that in gPC
method and there is no restriction on the dimensions of the random variables
neither.
