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Formula Used
An eigenvector v corresponding to eigenvalue λ satisfies A v = λ v. For a chosen λ, a nontrivial solution of (A−λI)v=0 spans the eigenspace (found via RREF). Normalization rescales vectors: v̂ = v / ||v||, using L2, L1, or L∞ norms.
Power iteration repeatedly applies A to a vector and normalizes, converging to the dominant eigenpair. Inverse iteration with shift σ solves (A−σI) y = v then normalizes, sharpening towards an eigenvector near σ. The Rayleigh quotient λ = (vᵀ A v)/(vᵀ v) refines the eigenvalue estimate. Residual is ||A v̂ − λ v̂||₂.
How to Use
- Select your matrix size, then enter matrix entries row-wise.
- Pick a method: RREF with a supplied eigenvalue, power iteration, or inverse iteration.
- Choose normalization, set tolerance, iterations, and decimals.
- Click Compute to obtain a normalized eigenvector and diagnostics.
- Download your results as CSV or PDF using the buttons provided.
Example Data
| Example | Matrix A | Method | λ (if used) | One normalized eigenvector |
|---|---|---|---|---|
| 1 | diag(2,3,4) | RREF | 3 | [0, 1, 0] |
| 2 | [[2,1],[1,2]] | Power | — | ≈ (1/√2)[1, 1] |
| 3 | [[4,1,0],[1,3,0],[0,0,2]] | Inverse (shift 4) | 4 | ≈ normalize([1,0,0]) |
FAQs
A normalized eigenvector has unit length with respect to a chosen norm, typically the Euclidean (L2) norm.
Normalization removes arbitrary scaling, making comparisons and residual checks meaningful and numerically stable.
The RREF method returns a basis; you can normalize each basis vector and choose any nonzero linear combination.
This calculator focuses on real arithmetic. Complex cases may appear as failure to converge or zero-only nullspace.
Accuracy depends on spectral gaps, tolerance, and iterations. The residual norm and Rayleigh quotient help validate outcomes.
It measures how close A v − λ v is to zero. Smaller values indicate better eigenpair approximations.
L2 is standard. L1 and L∞ can be useful in certain optimization or robustness contexts.