  In this paper we provide a matched pair decomposition of the space of
symmetric contravariant tensors $\mathfrak{T}\mathcal{Q}$. From this procedure
two complementary Lie subalgebras of $\mathfrak{T}\mathcal{Q}$ under
\textit{mutual} interaction arise. Introducing a lift operator, the matched
pair decomposition of the space of Hamiltonian vector fields is determined.
According to these realizations, Euler-Poincar\'{e} flows on such spaces are
decomposed into two subdynamics: one of which is the Euler--Poincar\'{e}
formulation of isentropic fluid flows, and the other one corresponds with
Euler--Poincar\'{e} equations on higher order contravariant tensors ($n\geq 2$)
