The Hessian

The Jacobian of the gradient: how the slopes change as you move.

Contour map. Click or drag anywhere to move the point. Dashed contours are below ground level.

Drag the green patch over the surface. Drag empty space to turn the view.

Surface Patch size ±0.50 Overlay the quadratic model f + ∇f·d + ½dᵀHd Slowly turn the view

H = J (\nabla f) = [\begin{matrix} \frac{\partial^{2} f}{\partial x^{2}} & \frac{\partial^{2} f}{\partial x \partial y} \\ \frac{\partial^{2} f}{\partial y \partial x} & \frac{\partial^{2} f}{\partial y^{2}} \end{matrix}] =

[\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]

λ₁ = 0, λ₂ = 0 SADDLE

Arrows: eigenvectors of H. Orange: curves up (λ > 0). Blue: curves down (λ < 0). Arrow length grows with |λ|.

The eigenvalue signs describe the local curvature anywhere on the surface. They classify a minimum, maximum or saddle point only where the slope is zero: at hilltops, at the bottoms of pits, and at the passes between them. Current slope:

Interactive version of an animation by Basem Rajjoub, from the LinkedIn post on the Hessian. The layout, colors and wording follow that original.