Computation Lab: Matrix Powers and Spectral Modes

An interactive lab for seeing how eigenvalues control repeated matrix powers and how different initial states split into spectral modes.

Keywords

computation, simulation, visualization, eigenvalues, matrix powers

1 Lab Goal

This lab helps you see one specific fact:

the long-term behavior of repeated matrix powers is controlled by the eigenvalues attached to the modes present in the initial state.

2 Math Question

How do the mixing strength and the starting vector affect:

• the eigenvalues
• the iterates \(P^k x_0\)
• the decay of the disagreement mode

3 Model or Setup

We use the symmetric two-state averaging matrix

\[ P = \begin{bmatrix} 1-\alpha & \alpha \\ \alpha & 1-\alpha \end{bmatrix}, \]

where \(\alpha \in [0,0.5]\).

Its eigenvalues are

\[ 1 \qquad \text{and} \qquad 1-2\alpha. \]

4 Parameters and Controls

• Mixing α
• Initial x-coordinate
• Initial y-coordinate
• Number of steps k

5 Code and Simulation

viewof alpha = Inputs.range([0, 0.5], {
  value: 0.2,
  step: 0.01,
  label: "Mixing α"
})

viewof x0a = Inputs.range([-2, 2], {
  value: 1,
  step: 0.05,
  label: "Initial x-coordinate"
})

viewof x0b = Inputs.range([-2, 2], {
  value: 0,
  step: 0.05,
  label: "Initial y-coordinate"
})

viewof steps = Inputs.range([0, 20], {
  value: 8,
  step: 1,
  label: "Number of steps k"
})

P = [[1 - alpha, alpha], [alpha, 1 - alpha]]
lambda1 = 1
lambda2 = 1 - 2 * alpha

applyP = (v) => ({
  x: P[0][0] * v.x + P[0][1] * v.y,
  y: P[1][0] * v.x + P[1][1] * v.y
})

initialState = ({x: x0a, y: x0b})

trajectory = {
  let state = ({...initialState})
  let rows = [{step: 0, x: state.x, y: state.y}]
  for (let i = 1; i <= steps; i++) {
    state = applyP(state)
    rows.push({step: i, x: state.x, y: state.y})
  }
  return rows
}

consensusValue = (x0a + x0b) / 2
modeConsensus = ({x: consensusValue, y: consensusValue})
modeDifference = ({x: (x0a - x0b) / 2, y: -(x0a - x0b) / 2})

Plot.plot({
  height: 420,
  x: {label: "step"},
  y: {label: "state value"},
  color: {legend: true},
  marks: [
    Plot.gridX({strokeOpacity: 0.15}),
    Plot.gridY({strokeOpacity: 0.15}),
    Plot.line(trajectory.map((d) => ({step: d.step, value: d.x, coord: "first"})), {x: "step", y: "value", stroke: "coord", strokeWidth: 3}),
    Plot.line(trajectory.map((d) => ({step: d.step, value: d.y, coord: "second"})), {x: "step", y: "value", stroke: "coord", strokeWidth: 3}),
    Plot.ruleY([consensusValue], {stroke: "#94a3b8", strokeDasharray: "6,3"})
  ]
})

md`
- Eigenvalues: **1** and **${lambda2.toFixed(3)}**
- Initial consensus mode: **(${modeConsensus.x.toFixed(3)}, ${modeConsensus.y.toFixed(3)})**
- Initial disagreement mode: **(${modeDifference.x.toFixed(3)}, ${modeDifference.y.toFixed(3)})**
- Current state after ${steps} steps: **(${trajectory[trajectory.length - 1].x.toFixed(3)}, ${trajectory[trajectory.length - 1].y.toFixed(3)})**
`

6 What To Observe

• The average of the two coordinates stays fixed because the eigenvalue 1 mode persists.
• The disagreement shrinks at a rate controlled by \(|1-2\alpha|\).
• Larger \(\alpha\) makes the second eigenvalue smaller in magnitude, so convergence is faster.
• If the initial state already lies on the consensus line, nothing changes except trivial persistence.

7 Interpretation

This lab is a visual proof of the mode picture:

• one eigenvector is the consensus direction
• the other measures disagreement
• repeated powers preserve one mode and damp the other

That is exactly why eigenvalues matter in consensus, diffusion, and graph-based smoothing.

8 Failure Modes and Numerical Cautions

• This is a symmetric 2x2 example, so it hides complications from defective or nonnormal matrices.
• In larger graph systems, interpreting the spectrum depends on which operator is chosen.
• A single dominant eigenvalue is not the whole story when multiple modes are close in magnitude.

9 Reproducibility Notes

• execution engine: Observable JS
• randomness: none
• libraries: Quarto OJS with Plot and Inputs
• render mode: interactive client-side

10 Extensions

• compare this averaging matrix with a nonsymmetric update matrix
• track the spectral gap as \(\alpha\) varies

11 Sources and Further Reading

• MIT 18.06SC: Lecture 22, Diagonalization and Powers of A - First pass - official source for the powers-of-a-matrix viewpoint. Checked 2026-04-24.
• MIT 18.06: Lecture 21, Eigenvalues and Eigenvectors - Second pass - good official reinforcement on eigen-modes. Checked 2026-04-24.
• A Comprehensive Survey on Spectral Clustering with Graph Structure Learning - Paper bridge - useful later, once you want to see how modal thinking scales to graph operators. Checked 2026-04-24.