# Quadratic Forms

Business I · Mathematics II · https://tryals.app/learn/business-i/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.

| Type | $Q(\mathbf{x})$ | Shape | Optimum |
| --- | --- | --- | --- |
| **Positive def.** | $> 0$ always | Bowl | Min ($H > 0$) |
| **Negative def.** | $< 0$ always | Dome | Max ($H < 0$) |
| **Indefinite** | Both signs | Saddle | None ($H$ mixed) |
| **Semidefinite** | Touches zero | Flat | Boundary case |

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

## Practice questions

6 of this lesson's 14 practice questions, with answers. The full set is in the app.

### 1. Order from most constrained to least constrained curvature.

**Answer:**

1. Positive definite (all directions curve up)
2. Positive semidefinite (up or flat, never down)
3. Indefinite (up in some, down in others)

**Why:** Definite is the strongest constraint (all directions agree); semidefinite allows flatness; indefinite allows opposite curvatures.

Page: https://tryals.app/practice/business-i/quadratic-forms/order-from-most-constrained-to-least-constrained-curvature

### 2. Match each definiteness type to its geometric shape.

**Answer:**

- Positive definite → Bowl (curves up in all directions)
- Negative definite → Dome (curves down in all directions)
- Indefinite → Saddle (up in some, down in others)
- Positive semidefinite → Trough (flat in one direction, up in others)

**Why:** Definiteness determines curvature geometry, bowls, domes, saddles, and the flat borderline cases.

Page: https://tryals.app/practice/business-i/quadratic-forms/match-each-definiteness-type-to-its-geometric-shape

### 3. Definiteness of a quadratic form and the second-order condition for an optimum are the same test. Which correspondence makes that true?

A. Definiteness measures the determinant, which the SOC also uses
B. The quadratic form approximates the function itself near the critical point
C. They are related but distinct: definiteness concerns the matrix and the SOC concerns the function
D. The Hessian's definiteness IS the local curvature the SOC tests

**Answer:** D. The Hessian's definiteness IS the local curvature the SOC tests

**Why:** The first-order term vanishes at a critical point, so the local shape is entirely the quadratic form built from the Hessian, definiteness *is* the curvature condition. Option 2 confuses the second-order term with the whole approximation, which also carries the function's value.

Page: https://tryals.app/practice/business-i/quadratic-forms/definiteness-of-a-quadratic-form-and-the-second-order-condition-for

### 4. A negative definite Hessian at a critical point identifies a maximum. Why does the sign pattern settle it rather than the determinant alone?

A. Because negative definiteness means it curves down every way
B. Because the Hessian is only defined at maxima
C. Because the determinant of a negative definite matrix is always negative
D. Because the first-order conditions already rule out a minimum

**Answer:** A. Because negative definiteness means it curves down every way

**Why:** The eigenvalues are curvatures along the principal axes, and their signs decide the classification. A determinant can be positive with two negative eigenvalues, which is why it is not enough on its own.

Page: https://tryals.app/practice/business-i/quadratic-forms/a-negative-definite-hessian-at-a-critical-point-identifies-a-maximum

### 5. Which are required for positive definiteness of a 2x2 symmetric matrix?

A. The trace is zero
B. $a_{11}a_{22} - a_{12}^2 > 0$ (second principal minor positive)
C. $a_{11} > 0$ (first principal minor positive)
D. Both eigenvalues are positive

**Answer:** B. $a_{11}a_{22} - a_{12}^2 > 0$ (second principal minor positive); C. $a_{11} > 0$ (first principal minor positive); D. Both eigenvalues are positive

**Why:** All three are equivalent conditions for positive definiteness. The trace being zero would mean the eigenvalues sum to zero, implying indefiniteness.

Page: https://tryals.app/practice/business-i/quadratic-forms/which-are-required-for-positive-definiteness-of-a-2x2-symmetric

### 6. A symmetric matrix with all positive eigenvalues is positive definite. Why does that matter for a second-order condition?

A. Because positive definiteness is required for the matrix to be invertible
B. Because it curves upward in every direction from the point
C. Because it guarantees the stationary point is unique
D. Because it ensures the first-order conditions have a solution

**Answer:** B. Because it curves upward in every direction from the point

**Why:** The eigenvalues are the curvatures along the principal axes. Mixed signs mean up one way and down another, a saddle, which is why the sign pattern, not the determinant alone, settles the classification.

Page: https://tryals.app/practice/business-i/quadratic-forms/a-symmetric-matrix-with-all-positive-eigenvalues-is-positive