I have a question about the following excerpt from the paper("An Iterative Minimization Formulation for Saddle-Point Search") by Gao,Leng, Zhou on gentlest ascent in dynamical systems.
I am having a hard time understanding the part where it says that "the equation (1b) attempts to find the direction that corresponds to the smallest eigenvalue of the Hessian $\nabla ^2 V(x)$"
If I drop the normalization term in (1b) and set the relaxation constant $\gamma = 1$, then I am basically looking at
$$\dot v = - \nabla ^2 V(x)v \tag A$$, but why is it that this dynamic attempts to find the direction corresponding to the smallest eigenvalue of the Hessian of V?
For a fixed $x$ denote $A:=\nabla^2 V(x)$. It is a matrix. Then locally your equation looks like $\dot{v} = -A v$. By standard analysis of linear ODE if $-A$ has all e.v. not laying on the imaginary axis, then the dynamics will follow the largest e.v. of $-A$ i.e. minus the smallest e.v. of $A$.