Matrix Input
Enter a real square matrix (2×2 to 6×6). Leave blank cells as 0.
Example Data Table
This symmetric matrix has real eigenvalues and orthogonal eigenvectors.
| 4 | 1 | 1 |
| 1 | 3 | 0 |
| 1 | 0 | 2 |
Formula Used
An eigenvector v ≠ 0 and eigenvalue λ satisfy: A v = λ v. Rearranging gives: (A − λI) v = 0.
Eigenvalues are approximated using QR iteration on A. When the off-diagonal norm is small, diagonal entries approximate eigenvalues.
For each real λ, an eigenvector comes from a nullspace vector of A − λI computed by row-reduction. A residual check reports ‖A v − λ v‖₂.
How to Use This Calculator
- Select matrix size and enter entries. Blanks become zeros.
- Adjust iterations and tolerance for harder matrices.
- Enable normalization to compare directions consistently.
- Press Submit. Results show above the form.
- Export CSV or PDF to save eigenpairs and checks.
Why Eigenvectors Matter
Eigenvectors capture directions a linear transformation keeps aligned. In mechanics, they represent mode shapes; paired eigenvalues connect to frequency or stability. In data work, covariance eigenvectors create principal components that compress features while retaining variance. In networks, Laplacian eigenvectors expose communities and diffusion speed. Reliable eigenvectors turn raw coefficients into interpretable structure. For control systems, dominant eigenvectors indicate states that grow fastest, guiding sensor placement, damping strategies, and robust feedback tuning decisions overall.
What the Calculator Computes
The tool estimates eigenvalues with QR iteration until off‑diagonal energy is small. For each real eigenvalue λ, it solves (A−λI)v=0 by row reduction to find a nonzero nullspace vector. Optional normalization scales v to unit length, keeping direction unchanged. A residual norm reports how closely Av matches λv.
Reading the Residual Norm
Residuals quantify accuracy: r=Av−λv and ‖r‖₂. Values near zero indicate the eigenpair satisfies the equation within numerical limits. Larger residuals can signal insufficient iterations, tight eigenvalue clustering, or ill‑conditioned matrices. Comparing residuals across eigenpairs helps identify which modes are dependable for design, prediction, or reporting.
Iterations, Tolerance, and Shifting
Higher iteration limits usually improve convergence, but cost time. Tolerance controls the stopping rule; smaller tolerances seek a cleaner upper‑triangular form. A simple shift can accelerate convergence by moving the spectrum closer to the diagonal during QR steps. If results fluctuate, increase iterations, enable shifting, and recheck residual trends.
Best Cases and Hard Cases
Real symmetric matrices typically yield real eigenvalues and orthogonal eigenvectors, making interpretation stable and plots easy to read. Non‑symmetric matrices may produce complex conjugate pairs; those are flagged because this calculator reports real eigenvectors only. Repeated or defective eigenvalues can produce unstable directions; residual checks become especially important there.
Exporting and Reproducible Use
CSV export supports spreadsheets, notebooks, and pipelines, while PDF export produces a portable report. Save size, tolerance, iterations, sorting, and normalization settings alongside eigenpairs so colleagues can reproduce the computation. When comparing revisions of a model, track changes in eigenvalues and residuals to confirm improvements are real, not numerical noise.
FAQs
1) Why do I see “complex pair” in the results?
Some real matrices have complex eigenvalues. This tool focuses on real eigenvectors, so complex conjugate pairs are marked and excluded from vector output.
2) What does “Normalize eigenvectors” change?
Normalization scales each eigenvector to unit length. It does not change direction, but makes magnitudes comparable across eigenpairs and improves plotting readability.
3) How should I judge if an eigenpair is accurate?
Use the residual norm ‖Av−λv‖₂. Smaller values indicate the computed vector closely satisfies the eigen equation for the displayed eigenvalue.
4) Why might residuals stay large even with many iterations?
Ill-conditioned matrices, repeated eigenvalues, or near-defective structures can make eigenvectors unstable. Try a tighter tolerance, different scaling, or a symmetric reformulation.
5) Does sorting eigenvalues affect eigenvectors?
Sorting only reorders the reported eigenvalues. Eigenvectors are recomputed for the displayed order, but the underlying directions remain associated with their eigenvalues.
6) What matrix sizes are supported here?
The interface supports 2×2 through 6×6. Larger matrices are possible in principle, but would need more robust numerical routines and performance tuning.