Hessian Matrix — Interactive Visualizer

Surface Function

Hover the cursor over the surface.
▶ Yellow arrow = gradient (steepest ascent).
▶ Teal plane = tangent plane.

Warm colors = positive height.
Cool colors = negative height.

Hovered Point

Move over the surface to inspect the local point.

Jacobian J(x, z)

[

--0.00 0.00--

]

Hover Hessian H(x, z)

[

---- ----

]

Hessian Entries

f_xx = 1.00

f_yy = 1.00

f_xy = 0.00

Hessian Matrix H

[

1.000.00 0.001.00

]

Positive Definite ▲

Ball Simulation

Drag f_xy to twist the surface diagonally — like wringing a towel.

The ball rolls toward the minimum when positive definite, and escapes when indefinite.

Hessian Entries

f_xx = 1.00

f_yy = 1.00

f_xy = 0.00

Principal Curvatures

λ₁ (max curvature) 1.000

λ₂ (min curvature) 1.000

The ellipse is the level curve $f(x, z) = 2$.
Arrows = eigenvectors (principal axes).
Warm arrow = larger λ Cool arrow = smaller λ.

For an indefinite matrix, the level curve becomes a hyperbola, and both branches are drawn.

Ball Simulation

Eigenvalues at Origin

For $y = 0.4(x^2 - z^2)$:

λ₁ = +0.8 along x — stable

λ₂ = −0.8 along z — unstable

The ball returns along the green axis (bowl-shaped) and escapes along the red axis (inverted-bowl-shaped).

This is why Hessian eigenvalues classify critical points.