Calculator Input
Example Data Table
| Dimension | Matrix A | Eigenvalue λ | Input Vector v | L2 Norm | Normalized Eigenvector |
|---|---|---|---|---|---|
| 3 | [[2, 1, 0], [1, 2, 0], [0, 0, 3]] | 3 | [1, 1, 0] | 1.414214 | [0.707107, 0.707107, 0] |
Formula Used
1. Start with a nonzero eigenvector v = [v₁, v₂, ..., vₙ].
2. Compute the Euclidean norm:
||v||₂ = √(v₁² + v₂² + ... + vₙ²)
3. Divide every component by that norm:
u = v / ||v||₂
4. The normalized eigenvector u has unit length, so ||u||₂ = 1.
5. If a matrix and eigenvalue are supplied, the check uses r = Av - λv. A very small residual indicates the vector matches the eigenvalue relation well.
How to Use This Calculator
- Choose the vector dimension from 2 through 6.
- Enter every component of the eigenvector.
- Pick the decimal precision for displayed results.
- Enable matrix verification only when you also know the matrix and eigenvalue.
- Click the calculate button to show the normalized eigenvector above the form.
- Review norms, ratios, the optional residual, and the chart.
- Use the download buttons to save CSV or PDF output.
How to Normalize Eigenvectors
Find the vector’s Euclidean norm by squaring each component, adding the squares, and taking the square root. Then divide every component by that norm. The result points in the same direction, but its length becomes exactly one.
Frequently Asked Questions
1) What does this calculator return?
It returns the unit version of the entered eigenvector, along with its original norms, direction ratios, and an optional residual check against a supplied matrix and eigenvalue.
2) How to normalize eigenvectors?
Compute the Euclidean norm of the eigenvector, then divide each component by that norm. The resulting vector has length one and still represents the same eigendirection.
3) Why does normalization not change the eigendirection?
Scaling a nonzero eigenvector by any nonzero constant changes its length only. Its direction stays the same, so it still represents the same eigenvector line.
4) Can the zero vector be normalized?
No. The zero vector has norm zero, so division by the norm is impossible. Eigenvectors must always be nonzero.
5) What does the residual Av − λv tell me?
It measures how closely the supplied vector and eigenvalue satisfy the eigenvalue equation. Smaller residuals mean the input behaves more like a true eigenpair.
6) Why are signs sometimes different between answers?
If u is a normalized eigenvector, then −u is also normalized and valid. Both point along the same eigendirection, only with opposite orientation.
7) Should I use L2, L1, or infinity norm here?
Standard eigenvector normalization usually uses the Euclidean L2 norm. The page also shows L1 and infinity norms for comparison and interpretation.
8) When is normalization especially useful?
Normalization is useful in linear algebra proofs, numerical algorithms, quantum mechanics, PCA, and exam work because unit vectors simplify comparisons and computations.