Eigenvalues and Eigenvectors

Numerical methods for computing eigenvalues and eigenvectors of matrices.

Power Method

Iterative method for finding the dominant eigenvalue and corresponding eigenvector.

Interactive Demonstration

Matrix A (3x3)

Initial Vector v(0)

Max Iterations

Practice Problems

Use the power method to find the dominant eigenvalue of each matrix.

Problem 1

Matrix A:

[4, 1]

[2, 3]

Initial v(0):

[1, 1]

Dominant eigenvalue

Problem 2

Matrix A:

[3, -1]

[1, 1]

Initial v(0):

[1, 0]

Dominant eigenvalue

Problem 3

Matrix A:

[5, 2]

[1, 4]

Initial v(0):

[1, 1]

Dominant eigenvalue

Implementation

INPUT: A, x0, TOL, max_iter
OUTPUT: dominant eigenvalue λ, eigenvector x

x = x0 / ||x0||
for k = 1, 2, ..., max_iter:
    y = A * x
    λ = x^T * y
    x_new = y / ||y||
    if ||x_new - x|| < TOL:
        RETURN λ, x_new
    x = x_new

RETURN "Method failed"