Define function $f$: $\mathbb R^N\to \mathbb R$ by $$f(\xi):=A\xi\cdot\xi $$ where $A$ is $N \times N$ uniformly elliptic matrix, i.e., $A\xi\cdot\xi\geq \theta\lvert\xi\lvert^2$ for some $\theta>0$.
My question is as following:
$(1)$: is function $f$ convex, or even strict convex? I think yes, and I show it by proving function $$ a(t):=f(\xi+t\xi_2) $$ is convex at $0$ by computing second derivative. But I was wondering is there a better explanation by computing from vector respect?
$(2)$: What is $$ \nabla f(\xi)? $$ I think it is $2A\xi$ but I can only show it by using brute force calculation, i.e., write done every element of $A\xi\cdot\xi$ and do $1-D$ calculation and guess what is should be in the vector form... Again...Is there a quicker way to do the derivative?