1. Power Iteration — Dominant Eigenvector

Visualizing vector convergence to the dominant eigenvector

Formulation

x_{k+1} = \\frac{A x_k}{\\|A x_k\\|}

By repeatedly multiplying a vector by $A$ and normalizing, the vector is increasingly dominated by the component along the eigenvector with the largest absolute eigenvalue.

Geometric Model

A = \\begin{bmatrix} 2 & 1 \\\\ 0 & 1 \\end{bmatrix}

Eigenvalues: $\\lambda_1 = 2, \\lambda_2 = 1$

Eigenvector $p_1$ (red) corresponds to $\\lambda_1 = 2$ .
Eigenvector $p_2$ (red) corresponds to $\\lambda_2 = 1$ .
Notice how the component along $p_1$ grows twice as fast as the component along $p_2$ at each step.

Controls

θInitial Vector

x_0

45°

Iteration: 0