  Diffeomorphisms can be seen as automorphisms of the algebra of functions. In
the matrix regularization, functions on a smooth compact manifold are mapped to
finite size matrices. We consider how diffeomorphisms act on the configuration
space of the matrices through the matrix regularization. For the case of the
fuzzy $S^2$, we construct the matrix regularization in terms of the
Berezin-Toeplitz quantization. By using this quantization map, we define
diffeomorphisms on the space of matrices. We explicitly construct the matrix
version of holomorphic diffeomorphisms on $S^2$. We also propose three methods
of constructing approximate invariants on the fuzzy $S^2$. These invariants are
exactly invariant under area-preserving diffeomorphisms and only approximately
invariant (i.e. invariant in the large-$N$ limit) under the general
diffeomorphisms.
