Quadratic Forms

The Shape of Curvature

A quadratic form in $n$ variables is a function $Q(\mathbf{x}) = \mathbf{x}^T A \mathbf{x}$, where $A$ is a symmetric matrix. 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 shape:

• Positive definite ($Q > 0$ for all $\mathbf{x} \neq \mathbf{0}$): a bowl curving upward in every direction. Local minimum.
• Negative definite ($Q < 0$ for all $\mathbf{x} \neq \mathbf{0}$): a dome curving downward. Local maximum.
• Indefinite ($Q$ takes both signs): a saddle. Neither max nor min.
• Semidefinite ($Q \geq 0$ or $Q \leq 0$, touching zero): boundary case — flat in some direction.

Principal-minor test (Sylvester's criterion) — check the leading principal minors $\Delta_k$ of $A$:

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

For a 2x2 matrix:

• Positive definite: $\Delta_1 > 0$ and $\Delta_2 > 0$.
• Negative definite: $\Delta_1 < 0$ and $\Delta_2 > 0$ (alternating signs).
• If neither pattern holds: indefinite or semidefinite.

Connection to optimisation: the Hessian matrix $H$ at a critical point is exactly the matrix of a quadratic form (the second-order Taylor approximation). The definiteness of $H$ determines the nature of the critical point:

$$H \text{ positive definite} \Rightarrow \text{local minimum}$$ $$H \text{ negative definite} \Rightarrow \text{local maximum}$$ $$H \text{ indefinite} \Rightarrow \text{saddle point}$$

This is a more general version of the Hessian test from Lesson 3, extended to any number of variables.

Definiteness at a glance

Tip: Sylvester's criterion in two variables is quick: $\Delta_1 = a_{11}$ and $\Delta_2 = \det A$. Both positive → positive definite; signs alternating ($\Delta_1 < 0$, $\Delta_2 > 0$) → negative definite. Anything else, check further.