Hessian matrices can be used in the second partial derivative test to determine whether a stationary point is a maximum, a minimum, or a saddle.
Can Hessians also be used to gain insight into non-stationary points? I.e., if a random point in a high-dimensional space is taken, and the Hessian of a function is calculated for that point, would Hessian capture local curvature information? Would the eigenvalues still convey a similar meaning?