The Hessian
Let H ∈ M_{2}(ℝ) be symmetric (by Clairaut), Q(x)=xᵀHx
left: z = Q(x)
right: Q(θ) = h(θ)ᵀHh(θ), h(θ) = (cosθ, sinθ)
i.e. restriction of Q to the unit circle (curvature by direction)
classification:
eigenvectors = principal directions
eigenvalues = values of Q along them
more: MathNotes
u/CantorClosure — 6 days ago