Overview
This lecture introduced the Rayleigh Quotient method as a faster way to compute eigenvalues and eigenvectors, compared its efficiency to previous iterative methods, and provided details on the upcoming midterm exam.
Review of Previous Methods
- Power iteration finds the largest eigenvalue and its eigenvector through repeated matrix-vector multiplication.
- Normalized power iteration normalizes vectors at each step to keep values manageable.
- Inverse power iteration finds the smallest eigenvalue.
Rayleigh Quotient Method
- The Rayleigh Quotient estimates the eigenvalue given an approximate eigenvector for matrix A.
- Formula: λ = (xᵗAx) / (xᵗx), where x is the approximate eigenvector.
- This method treats the eigenvalue problem as a least squares problem, using normal equations.
- Rayleigh Quotient accelerates convergence compared to power iteration methods.
Example: Application to a 2x2 Matrix
- Matrix A = [[3, 1], [1, 3]], initial vector x₀ = [0, 1].
- In each iteration: compute yₖ = A·xₖ₋₁, normalize xₖ = yₖ / ||yₖ||_∞, update λₖ = (xₖᵗA xₖ)/(xₖᵗxₖ).
- Table showed Rayleigh Quotient converges to the dominant eigenvalue (4) in about 6 steps, faster than normalized power iteration.
Comparison to Previous Methods
- Rayleigh Quotient converges in fewer steps (about 6) compared to normalized power iteration (about 12 steps for same accuracy).
- Quotient value at each step is larger and approaches the true eigenvalue faster.
Exam Information
- Midterm exam is next class (Week 10), covering all material up to Lecture 13.
- Exam types: short answer (concepts/definitions), true or false (properties), computation (problems using learned methods).
- Computation questions will focus on vector norms, matrix computations, elementary elimination, least squares, and eigenvalue/eigenvector methods.
- Exam duration is 90 minutes; to be submitted via Blackboard.
Key Terms & Definitions
- Eigenvalue — a scalar λ such that Ax = λx for some nonzero vector x.
- Eigenvector — a nonzero vector x satisfying Ax = λx.
- Rayleigh Quotient — λ = (xᵗAx) / (xᵗx), an estimate of the eigenvalue for a given vector.
- Infinity Norm (||x||_ ∞) — the maximum absolute value among the elements of vector x.
- Least Squares Problem — minimizes the sum of squared residuals; used here to estimate eigenvalues.
Action Items / Next Steps
- Review lecture slides, homework, and labs—especially examples and computations.
- Practice applying Rayleigh Quotient and other eigenvalue methods.
- Prepare for the midterm by studying key concepts, definitions, and calculation steps.