Let $L$ be an unbounded operator, such that
$$L = \operatorname{div} \left( M \nabla_{x}\right),$$
where $M$ is a non-degenerate symmetric $n\times n$-matrix ($ \det M\neq 0$) and $\nabla_{x}$ is the gradient vector on $\mathbb R^n$, i.e.,
$$\nabla_{x}= \left(\frac{\partial }{\partial x_{1}},\dots,\frac{\partial }{\partial x_{n}}\right)^t .$$
What can we say about the ellipticity of the operator $L$?
Thanks