We consider how diffeomorphisms act on the configuration space of matrices in the
matrix regularization. We construct the matrix regularization in terms of the Berezin-Toeplitz quantization and define diffeomorphisms on the space of matrices by using this quantization map. For the case of the fuzzy $S^2$, 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 are exactly invariant under unitary similarity transformations and only approximately invariant under the general diffeomorphisms.