For a gradient system $\dot{x}=-\nabla V(x)$, $V$ is $C^2$, prove the eigenvalues of the derivative of the vector field at every fixed point are real

Question

I want to show that for a gradient system $\dot{x}=-\nabla V(x)$, where $V$ is $C^2$, the eigenvalues of the derivative of the vector field at every fixed point are real.

I know that if $x_0$ is a fixed point then $\nabla V(x_0)=0$, but have no idea where to go from this.

Answer 1

The derivative of the vector field is the Hessean of the potential. For a $C^2$ function that is a symmetric matrix, now apply the spectral theory of symmetric or self-adjoint matrices.