Practice question · Multiple choice
A symmetric matrix with all positive eigenvalues is positive definite. Why does that matter for a second-order condition?
Hints
- Ask what a saddle point looks like along two different directions.
- The quadratic form is the curvature in an arbitrary direction.
Show the 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.
Practise Quadratic Forms
The app has 8 more questions on this lesson, and keeps your place in the course. Business I is free to start.
More questions on Quadratic Forms
- Order from most constrained to least constrained curvature.
- Match each definiteness type to its geometric shape.
- Definiteness of a quadratic form and the second-order condition for an optimum are the same test. Which…
- A negative definite Hessian at a critical point identifies a maximum. Why does the sign pattern settle it…
- Which are required for positive definiteness of a 2x2 symmetric matrix?