Quadratic Form on a Constraint Set¶

Definition¶

Let $q(\boldsymbol{h})=\boldsymbol{h}^{\top} \boldsymbol{A}\boldsymbol{h}$ be a quadratic form and let a differentiable equality constraint system $\boldsymbol{g}(\boldsymbol{x})=\boldsymbol{0}$ have linearized direction space

T=\{\boldsymbol{h}\in\mathbb{R}^n\colon\boldsymbol{D}\boldsymbol{g}(\boldsymbol{a})\boldsymbol{h}=0\}.

The quadratic form is positive definite on the constraint set at $\boldsymbol{a}$ (that is, on the constraint directions $T$ , not on the feasible set $\{\boldsymbol{g}=\boldsymbol{0}\}$ itself) if

\boldsymbol{h}^{\top} \boldsymbol{A}\boldsymbol{h}>0

for every nonzero $\boldsymbol{h}\in T$ $T$ . Negative definiteness on the constraint set is defined similarly with $\boldsymbol{h}^{\top} \boldsymbol{A}\boldsymbol{h}<0$ $0$ .

Interpretation¶

Under constraints, curvature only matters in directions that are feasible to first order. The same matrix can look indefinite overall but decisive on the allowed directions.

Level curves of an indefinite quadratic form in the plane, positive in one pair of sectors and negative in the other, with a line T through the origin on which the form is positive.

The form $q(\boldsymbol{h})=\boldsymbol{h}^{\top} \boldsymbol{A}\boldsymbol{h}$ is indefinite on the full space but positive definite on the tangent line $T$ drawn. A constrained second-order test examines this restricted curvature rather than every direction in the ambient space.

An independent linear constraint system identifies the linearized directions left free by the constraints. The test considers only its null space: curvature in a forbidden direction cannot decide whether the objective improves along feasible motion.

Key Results¶

Under the full-row-rank regularity condition, the kernel $T$ is the tangent space of the equality constraint set. Active inequality constraints generally lead to a cone of feasible directions instead.
Constrained second-order tests study the Hessian of the Lagrangian on the feasible direction space.
Bordered matrix tests are determinant-based ways to check this restricted definiteness.

Examples¶

For $q(h_1,h_2)=h_1^2-h_2^2$ with constraint $h_2=0$ $0$ , the restricted form is positive definite.
For $q(h_1,h_2)=h_1^2-h_2^2$ with constraint $h_1=0$ $0$ , the restricted form is negative definite.

Prerequisites¶

Sources¶

Math IV: Multivariate Calculus · Week 3, Lecture 2

Aliases: constrained quadratic form Keywords: restricted quadratic form, quadratic form on tangent space, definiteness under constraints Symbols: element of, real numbers

Last updated: 17 October 2026 21:53