Eigenvalues of a second derivative

Question

I have a function f(r) that describes a Gaussian random field. A second derivative can be formed $\nabla_i \nabla_j f(r)$. I am looking at a paper that claims that in finding the extremum, the eigenvalues of the second derivative operator should be negative definite. I was curious whether anyone knew where this rule came from? I've never seen anything written up about this particular second derivative except in the context of Riemann tensors. Any insight would be appreciated!

Answer 1

A matrix $A$ is negative definite if $$\forall x\ne0,\quad\langle x,Ax\rangle<0$$ hence in a critical point of $f$ i.e. $\nabla f(a)=0$ and if the Hessian matrix is definite negative then:

\begin{align} f(x) = f(a)+\underbrace{\nabla f(a)}_{=0} \cdot (x-a) + \frac{1}{2}\underbrace{(x-a)^T \mathbb{H}(a) (x-a)}_{<0}+ o(||x-a||^{2}). \end{align} hence we see that $f(x)<f(a)$ in a neighberhood of $a$ so $f(a)$ is a local minimum of $f$.