Shapes of Quadratic Functions

In two dimensions, the graph of a quadratic form $z = f(x_1, x_2) = \lambda_1 x_1^2 + \lambda_2 x_2^2$ takes on beautifully distinct shapes depending on the signs of $\lambda_1$ and $\lambda_2$. When both are positive, the graph is a bowl opening upward (elliptic paraboloid) — it has a unique global minimum at the origin. When both are negative, the bowl flips to open downward (unique global maximum). When $\lambda_1 > 0$ and $\lambda_2 < 0$, the graph is a saddle — upward in one direction, downward in another, with no global minimum or maximum.

What if one eigenvalue is zero? When $\lambda_1 > 0$ and $\lambda_2 = 0$, the surface is a half-pipe or parabolic cylinder — it curves up in the $x_1$-direction but is flat in the $x_2$-direction. Points along the $x_2$-axis are all global minima (not a unique minimum). This geometric classification based on eigenvalue signs is exactly what definiteness formalizes (Section 8.16).