Here's the problem:
Let $A$ be a positive definite symmetric matrix and let $Q(\mathbf x)$ denote the associated quadratic form on $\mathbb R^n$. Show that $f(\mathbf x)=e^{Q(\mathbf x)}$ is a convex function.
Here's what I know:
I know that if the Hessian of some function is positive definite, then that function is strictly convex. I also know that, obviously, $f(x) = e^x$ is convex, so I would need only to show that $Q(\mathbf x)$ is convex. I don't know of any thing that says if $A$ is positive definite and symmetric, then its associated quadratic form is convex, but perhaps that's true? If so, how would I go about showing it?
