Power Method Eigenvalue Calculator

Iterate toward the leading eigenpair with transparent steps, normalization controls, and stopping rules for reliability. Input any square matrix, set an initial vector, and watch convergence diagnostics update live interactively. See Rayleigh quotients, residual norms, and eigenvector scaling across iterations for insight. Export tables as CSV, generate quick PDFs, and document every computational step.

Shift σ
Inputs
Tip: Enter numbers like 3, -1.25, or scientific 6.2e-3.
Results —
Eigenvalue estimate (Rayleigh μ)
—
Iterations used
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Eigenvector (normalized)
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Step-by-step Log
# Rayleigh μ ‖residual‖ scale x₁ x₂ x₃ x₄…
Residual: ‖A x − μ x‖ using selected normalization for the iterate scaling.
Convergence of Rayleigh Quotient
Tracks μ per iteration. Multiple runs overlay as separate traces.
Residual Norm per Iteration (log₁₀)
Lower is better; flat lines may indicate stagnation or ill-conditioning.
Example Data Table
Example A (3×3)
410
131
012
Example initial vector: [1, 1, 1]^T
Formulas Used
  1. Power: $x_{k+1} = \\dfrac{A x_k}{\\|A x_k\\|}$ with chosen vector norm.
  2. Inverse (shifted): solve $(A-\\sigma I) y_k = x_k$, set $x_{k+1} = \\dfrac{y_k}{\\|y_k\\|}$.
  3. Rayleigh: $\\mu_k = \\dfrac{x_k^{\\top} A x_k}{x_k^{\\top} x_k}$ estimates the associated eigenvalue.
  4. Residual: $r_k = A x_k - \\mu_k x_k$; stop when $\\|r_k\\|$ or $|\\mu_k-\\mu_{k-1}|$ is below tolerance.
Inverse iteration targets the eigenvalue nearest the shift $\\sigma$ when conditions permit.
How to Use
  1. Set n, then enter a square matrix.
  2. Provide an initial vector or enable random initialization.
  3. Choose tolerance, iteration cap, and normalization type.
  4. Click Run Power Method for dominant eigenpair estimates.
  5. To seek an eigenpair near a target value, set shift σ and click Run Inverse (Shifted).
  6. Review logs, charts, and exports to document convergence behavior thoroughly.
FAQs

It estimates an eigenvalue–eigenvector pair: either the dominant pair or the one closest to a given shift via inverse iteration.

Power converges when a unique largest-magnitude eigenvalue exists. Inverse iteration converges near the eigenvalue closest to your shift, assuming suitable conditioning.

Use prior knowledge, Gershgorin bounds, or experiment by scanning values. Closer shifts yield faster convergence to the desired eigenpair.

Near-singularity can stall or explode iterations. Nudge σ slightly, regularize, or try another starting vector to avoid exact singularity.

It provides a consistent eigenvalue estimate from any iterate and typically converges faster than the raw scaling factors.

Yes, run inverse iteration with different shifts or apply deflation strategies after finding one eigenpair.

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