Quadratic Forms

The Shape of Curvature

A quadratic form is $Q(\mathbf{x}) = \mathbf{x}^T A \mathbf{x}$ with symmetric $A$. In two variables:

$$Q(x_1, x_2) = a_{11}x_1^2 + 2a_{12}x_1 x_2 + a_{22}x_2^2$$

Definiteness classifies the geometric shape and critical points. The principal-minor test checks leading principal minors $\Delta_k$:

$$\Delta_1 = a_{11}, \qquad \Delta_2 = a_{11}a_{22} - a_{12}^2$$

For $2 \times 2$ matrices:

• Positive definite: $\Delta_1 > 0$ and $\Delta_2 > 0$.
• Negative definite: $\Delta_1 < 0$ and $\Delta_2 > 0$.
• Indefinite: $\Delta_2 < 0$.
• Semidefinite: $\Delta_2 = 0$.

Definiteness at a Glance

The Hessian matrix $H$ at a critical point acts as our quadratic form. Its definiteness classifies local extrema.

Tip: Sylvester's criterion gives a quick test. Alternating signs ($\Delta_1 < 0, \Delta_2 > 0$) means negative definite; both positive means positive definite.