Given a vector $u=(x,y,z)\in \mathbb R^3$ and a $3\times 3$ real matrix $M$, I woud like to know if there exists some formulas to express in other manner the two quantities: $\nabla\nabla\cdot(Mu)$ and $\nabla\times(Mu)$ in terms of $M$ and $u$.
Also, I want to know for which type of matrix $M$ we have
$$
\nabla\nabla\cdot(Mu)=M \nabla\nabla\cdot u
$$
and
$$
\nabla\times(Mu)=M \nabla\times u.
$$
With summation over repeated indices, let's write vectors as $u=u_ke_k$, matrices as $M=M_{ij}e_{ij}$ etc. Then$$\nabla\cdot Mu=\partial_i(Mu)_i=M_{ij}\partial_i u_j$$(I'm assuming you want constant $M$), so$$\nabla\nabla\cdot Mu=\partial_k(M_{ij}\partial_iu_j)e_k=M_{ij}\partial_i\partial_ku_je_k.$$This can't be written as $Mv$ for a vector $v$, but it can be written as $M:v$ ($:$ denotes summation over each index of $M$) for a rank-3 tensor$$v:=\partial_i\partial_k u_j e_{ij}\otimes e_k,$$where $\otimes$ denotes a tensor product. Similarly,$$\nabla\times(Mu)=\epsilon_{ijk}M_{jl}\partial_i u_le_k,$$i.e. $M:v$ with $v:=\epsilon_{ijk}\partial_i u_le_{jl}\otimes e_k$.