2. Power Iteration — Spectral Decay

Visualizing vector decay when spectral radius ρ(A) < 1

Formulation

x_{k+1} = A x_k

When all eigenvalues of $A$ have absolute value less than 1 (i.e. spectral radius $\\rho(A) < 1$ ), repeatedly multiplying by $A$ causes the vector to decay toward the origin.

Geometric Model

A = \begin{bmatrix} 0.8 & 0.3 \\ 0 & 0.5 \end{bmatrix}

Eigenvalues: $\\lambda_1 = 0.8, \lambda_2 = 0.5$

Eigenvector $p_1$ (red) corresponds to $\\lambda_1 = 0.8$ .
Eigenvector $p_2$ (red) corresponds to $\\lambda_2 = 0.5$ .
Notice how both components decay, but the component along $p_2$ decays much faster, meaning the vector trajectory curls into the $p_1$ axis as it shrinks to zero.

Controls

θInitial Vector

x_0

45°

Iteration: 0