Jacobi eigenvalue calculator form
Use this solver for real symmetric matrices only. Larger tolerances run faster. Smaller tolerances usually give more precise diagonalization.
Example data table
This example uses a 3 × 3 symmetric matrix. The listed eigenvalues are rounded reference values for quick validation.
Example input matrix
| 4.000000 | 1.000000 | 1.000000 |
| 1.000000 | 3.000000 | 0.000000 |
| 1.000000 | 0.000000 | 2.000000 |
Expected eigenvalues
| Index | Value |
|---|---|
| 1 | 4.879385 |
| 2 | 2.652704 |
| 3 | 1.467911 |
Formula used
Pivot selection: choose the largest off-diagonal element apq in magnitude.
Rotation angle: θ = 0.5 × atan2(2apq, aqq - app).
Jacobi rotation: set c = cos(θ) and s = sin(θ), then apply a plane rotation that zeros apq and aqp.
Diagonal form: repeated rotations produce A ≈ VDVT, where D contains eigenvalues and columns of V are eigenvectors.
Stopping rule: stop when the off-diagonal norm falls below the selected tolerance or when maximum iterations are reached.
The calculator also reports orthogonality error, reconstruction error, and residual checks so you can judge numerical quality beyond raw eigenvalues.
How to use this calculator
- Choose a matrix size from 2 × 2 up to 6 × 6.
- Enter a real symmetric matrix. Matching mirrored entries are required.
- Set tolerance, iteration cap, and displayed precision.
- Pick ascending or descending eigenvalue sorting.
- Press Compute eigenvalues to solve the system.
- Review the summary, eigenvalues, eigenvectors, residuals, and iteration log above the form.
- Download CSV or PDF reports when you need to document the run.
FAQs
1. What matrices work with this calculator?
Use real symmetric matrices only. The Jacobi eigenvalue method relies on symmetric structure to produce orthogonal eigenvectors and stable diagonalization.
2. Why must a(i,j) equal a(j,i)?
That equality defines symmetry. Without it, Jacobi rotations no longer preserve the assumptions needed for guaranteed orthogonal eigenvectors and clean convergence behavior.
3. What does tolerance control?
Tolerance sets the stopping threshold for the remaining off-diagonal norm. Smaller values usually improve accuracy but may need more iterations.
4. What if the solver does not converge?
Increase the iteration limit, check matrix symmetry, or choose a slightly larger tolerance. Harder matrices can require more rotations before the off-diagonal terms shrink enough.
5. What do the residual values mean?
Each residual estimates how closely an eigenpair satisfies Av = λv. Smaller residuals indicate that the computed eigenvalue and eigenvector pair is numerically stronger.
6. Why compare determinant and product of eigenvalues?
For square matrices, the determinant equals the product of eigenvalues. Matching values provide a useful consistency check on the numerical decomposition.
7. Are the eigenvectors normalized?
Yes. The rotation matrix stays orthogonal, so the resulting eigenvector columns remain normalized apart from tiny floating-point rounding differences.
8. When is this method a good choice?
It is excellent for learning, validation, and small symmetric systems where transparent iterations and orthogonal eigenvectors matter more than maximum speed.