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Power Method (Eigenvalues)

Dominant eigenvalue and direction by repeated matrix-vector products

Parameters

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Show Trail

Controls

xⓘ

Calculated Values

λ (power method):
3.00;3.00;
λ₁ (exact):
3.00;3.00;
λ₂ (exact):
1.00;1.00;
Dominant |λ|:
3.00;3.00;

Examples

Symmetric matrix

[[2,1],[1,2]] has λ = 3 and 1.

  • lambda: 3.003.00
  • v1: 0.710.71
  • v2: 0.710.71

Visualization

Eigenvalues in Physical Systems

Linear stability, normal modes, and quantum Hamiltonians all lead to eigenvalue problems Av = λv. The power method extracts the eigenvalue of largest magnitude and its eigenvector by iterating v_{k+1} = A v_k / ‖A v_k‖.

If |λ₁| > |λ₂| and the initial vector has a component along the λ₁ eigenvector, the iteration amplifies that direction exponentially. The Rayleigh quotient vᵀAv / vᵀv converges to λ₁.

For symmetric matrices in physics, eigenvalues are real and orthogonal eigenvectors represent independent modes — e.g. coupled oscillators or molecular vibrations.

Limitations: if |λ₁| ≈ |λ₂|, convergence is slow. Complex eigenvalues need extensions. For full spectra, use QR or Jacobi methods in production code.

The animation shows the vector direction aligning with the dominant eigenvector as iterations proceed.

Key Concepts

  • Dominant λ has largest |λ|
  • Normalize each step to prevent overflow
  • Rayleigh quotient estimates λ
  • Convergence rate ~ |λ₂/λ₁|ᵏ
  • Eigenvectors = mode shapes

Real-World Applications

  • Principal mode of coupled oscillators
  • Google PageRank-style network ranking
  • Stability exponents from Jacobian
  • Quantum energy estimates in large bases

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Physics Equations

Eigenproblem:
Av=λvA v = \lambda v
Power step:
vk+1=Avk∥Avk∥v_{k+1} = \frac{A v_k}{\|A v_k\|}

Step-by-Step Solution

See how the main results are calculated.

1

Step 1: Matrix eigenproblem

Find the largest-magnitude eigenvalue and an eigenvector direction.

Equation:

A=(2112)A = \begin{pmatrix}2 & 1\\1 & 2\end{pmatrix}
2

Step 2: Power iteration

Repeatedly multiply by A and normalize.

Equation:

vk+1=Avk∥Avk∥v_{k+1} = \frac{A v_k}{\|A v_k\|}

Explanation:

Converges to dominant eigenvector if |λ₁| > |λ₂| and v₀ has a component along λ₁.

3

Step 3: Rayleigh quotient estimate

Calculation:

λ≈3.000000 after 4 steps\lambda \approx 3.000000 \text{ after 4 steps}

Result:

Eigenvector ≈ (0.7071, 0.7071)
4

Step 4: Compare to analytic eigenvalues

Calculation:

λ1=3.000000,λ2=1.000000\lambda_1 = 3.000000,\quad \lambda_2 = 1.000000

Result:

Dominant λ ≈ 3.000000, power method λ ≈ 3.000000

Frequently Asked Questions (FAQ)

Why normalize each step?

Prevents vector magnitude from growing without bound; only direction matters for the eigenvector.

Can it find the smallest eigenvalue?

Use inverse iteration on A⁻¹ (or shift A − σI) to target other modes.

Practice MCQs

  1. Power method converges to the eigenvalue with:
  2. Av = λv means:
  3. Slow convergence occurs when:
  4. Normal modes of coupled springs are:
  5. Rayleigh quotient estimates:

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These tools connect to the same physics concepts used in this calculator.

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