Calculator Input
Example Data Table
This sample matrix is symmetric, giving real eigenvalues and orthogonal eigenvectors.
| Row | Column 1 | Column 2 | Column 3 |
|---|---|---|---|
| 1 | 4 | 1 | 0 |
| 2 | 1 | 3 | 1 |
| 3 | 0 | 1 | 2 |
Formula Used
For a square matrix A, an eigenvector v and eigenvalue λ satisfy:
Av = λv
Rearrange to the homogeneous form:
(A − λI)v = 0
Nontrivial solutions exist when:
det(A − λI) = 0
For each real eigenvalue, the calculator builds an eigenvector from the null space of A − λI. It then checks quality using the residual norm:
‖Av − λv‖
When a direct null-space vector is numerically weak, the tool refines the direction through inverse iteration with the selected tolerance and iteration limit.
How to Use This Calculator
- Select a 2 × 2 or 3 × 3 square matrix.
- Enter all matrix coefficients in the input grid.
- Choose how you want the eigenvector normalized.
- Set precision, tolerance, and maximum iterations.
- Optionally enter a target eigenvalue to isolate one direction.
- Click Find Eigenvectors to generate the output.
- Review eigenvalues, eigenvectors, Rayleigh checks, and residuals.
- Use the CSV or PDF buttons to save the results.
FAQs
1. What does this eigenvector finder calculate?
It calculates real eigenvalues and their associated eigenvectors for 2 × 2 and 3 × 3 matrices. It also reports residual error and a Rayleigh quotient check for better numerical validation.
2. Why might no result appear for some matrices?
If the matrix has only complex eigenvalues, this page does not display them. The current implementation is designed for real-valued outputs, which are often enough for many classroom and engineering use cases.
3. What does normalization change?
Normalization changes only the scale of the eigenvector, not its direction. Unit normalization gives length one, while max-component normalization makes the largest absolute component equal to one.
4. What is the residual norm?
The residual norm measures how closely the computed vector satisfies Av = λv. Smaller residual values indicate a stronger numerical match between the reported eigenvalue and eigenvector.
5. Can I calculate just one eigenvector?
Yes. Enter a target eigenvalue and the calculator will return the nearest computed real eigenvalue and its eigenvector. That helps when you already know which direction you want.
6. Why is the Rayleigh check useful?
The Rayleigh quotient estimates the eigenvalue implied by the returned vector. When it closely matches the listed eigenvalue, the solution is usually numerically consistent and trustworthy.
7. Is this suitable for symmetric matrices?
Yes. Symmetric matrices are especially well behaved because they have real eigenvalues and orthogonal eigenvectors. This calculator is very effective for those common matrix types.
8. Can I export the results?
Yes. The results block includes CSV and PDF download options. CSV is useful for spreadsheets, while PDF is convenient for sharing, printing, or saving calculation records.