  For finding the numerical solution of operator equations in many applications
a decomposition in subspaces is needed. Therefore, it is necessary to extend
the known method of matrix representation to the utilization of fusion frames.
In this paper we investigate this representation of operators on a Hilbert
space $\Hil$ with Bessel fusion sequences, fusion frame and Riesz
decompositions. We will give the basic definitions. We will show some
structural results and give some examples. Furthermore, in the case of Riesz
decompositions, we prove that those functions are isomorphisms. Also, we want
to find the pseudo-inverse and the inverse (if there exists) of such matrix
representations. We are going to apply this idea to the Schatten $p$-class
operators. Finally, we show that tensors of fusion frame are frames in the
space of Hilbert-Schmidt operators.
