Enter Matrix Values
This page reports real eigenvectors for real eigenvalues. Complex eigenvalues are listed clearly when they appear.
Example Data Table
| Matrix | Eigenvalues | Eigenvectors | Note |
|---|---|---|---|
| [[4, 1], [2, 3]] | 5, 2 | [1, 1], [1, -2] | Distinct real eigenpairs. |
| [[2, 0, 0], [0, 3, 1], [0, 1, 3]] | 2, 4, 2 | [1,0,0], [0,1,1], [0,-1,1] | Useful for repeated-value checks. |
| [[6, 2, 0], [2, 6, 0], [0, 0, 1]] | 8, 4, 1 | [1,1,0], [1,-1,0], [0,0,1] | Orthogonal eigenvectors appear clearly. |
Formula Used
The calculator starts from the eigenvector condition A v = λ v. It then rewrites the problem as (A - λI)v = 0.
Eigenvalues come from the characteristic equation det(λI - A) = 0. For each real eigenvalue, the calculator row-reduces A - λI and extracts a null-space basis.
When vector normalization is enabled, each eigenvector is scaled with v / ||v||, where ||v|| = √(v₁² + v₂² + ... + vₙ²).
How to Use This Calculator
- Select a 2 × 2 or 3 × 3 matrix size.
- Enter each matrix coefficient into the visible fields.
- Set decimal precision and a numeric tolerance.
- Choose whether vectors should be normalized.
- Submit the form to display eigenvalues and eigenvectors above it.
- Use the export buttons to save results as CSV or PDF.
Frequently Asked Questions
1. What does this calculator actually find?
It finds eigenvalues first, then computes matching real eigenvectors for each real eigenvalue of the entered 2 × 2 or 3 × 3 matrix.
2. Why can one eigenvalue have several vectors?
Any nonzero scalar multiple of an eigenvector is also valid. Repeated eigenvalues may also produce more than one linearly independent eigenvector.
3. What is the role of tolerance?
Tolerance controls when near-zero values are treated as zero. It helps stabilize row reduction and repeated-root grouping for floating-point calculations.
4. Why normalize eigenvectors?
Normalization rescales each vector to length one. It makes comparison easier, improves readability, and helps when matrices are used in further numeric workflows.
5. Does the calculator support complex eigenvalues?
Yes, it reports complex eigenvalues clearly. This page focuses on real eigenvectors for real eigenvalues, so complex eigenvector components are not expanded here.
6. Can I use decimal entries?
Yes. You can enter integers, decimals, or negative numbers. Precision and tolerance settings help tune how the final numeric output is displayed.
7. What indicates a defective matrix?
A defective matrix has fewer independent eigenvectors than its size. Compare algebraic multiplicity with geometric multiplicity to spot that situation quickly.
8. What matrices work best here?
This tool is best for classroom, homework, review, and quick verification tasks involving 2 × 2 and 3 × 3 real matrices.