Lab 14: Stability, Ranking, and Iteration

Explore what happens when a matrix is applied again and again. Watch stable directions emerge, rankings settle, probabilities converge, and signals smooth out.

Big idea

A matrix iteration has the form xk+1 = A xk. After many steps, eigenvectors often become visible because Akv = λkv.

growthdecayoscillationrankingMarkov chainspower iteration

1. Two-direction iteration

Change the two eigenvalues of a diagonal matrix. One direction may grow while another decays.

2. Power iteration

Repeated multiplication plus normalization can find a dominant eigenvector. The matrix is symmetric: A = [[a,b],[b,d]].

3. Markov chain weather model

Adjust transition probabilities. The columns represent today's weather; the entries describe tomorrow's distribution.

4. PageRank-style ranking

Importance flows through links. Damping adds a chance of jumping randomly to any page.

Columns are source pages; rows are destination pages. The rank vector is the stable distribution of importance.

5. Iteration as smoothing

Repeated local averaging removes sharp details. This is a simple model of diffusion.

Reflection

Use the interactive tools above to answer these questions:

  1. When does iteration converge to zero?
  2. When does it explode?
  3. Why does normalization help power iteration?
  4. What is the difference between one-step movement and long-run behavior?
  5. How are ranking and Markov chains both eigenvector problems?