Chapter 10 · Test 2
Advanced Linear Algebra
Vector spaces, eigenvalues, eigenvectors, and quadratic forms.
What you will learn
- Read eigenvalues of a triangular matrix
- Classify a quadratic form
- Verify an eigenvector
Reasoning habits that help
- For any triangular matrix, the characteristic polynomial factors using the diagonal entries.
- Definiteness classifies a quadratic form's sign in every nonzero direction and supports maximum, minimum, and saddle-point tests.
- An eigenvector keeps its direction under the matrix transformation; only its scale changes.
Where economists use these methods
- Eigenvalues describe long-run growth, stability, and convergence in dynamic economic systems and Markov models.
- Definiteness classifies curvature in utility, production, cost, and optimization problems.
- Eigenvectors identify persistent directions or steady compositions in growth, transition, and network models.