Let $A : \mathbb{R}^d \to \mathbb{R}^{d\times d}$. People say (http://www.numdam.org/item/COCV_2009__15_3_712_0/ (equation 1.3)) $A$ is uniformly elliptic if there exists $C_1,C_2>0$ such that for all $\xi\in \mathbb{R}^d$
$$ C_1\|\xi\|^2 \leq \langle A(x) \xi , \xi \rangle \leq C_2 \|\xi\|^2. $$
Some questions :
What is the reason for the name elliptic? Is it uniform because the $C_1,C_2$ hold for all $\xi$ or because they hold for all $x$? What do elliptic operators mean for the dynamics?