$L$ is an elliptic operator?

Question

Let $L$ be an unbounded operator, such that $$L = \sum_{i,j=1}^{n} h^{i,j}(x) \, \partial_{x_i} \partial_{x_j}.$$ If the matrix $h^{i,j}(x)$ is non-degenerate (or $ \det h^{i,j}(x) \neq 0$), is that we can say that $L$ is an elliptic operator ? If yes, any reference. Thanks.