Formula Used
An eigenpair (lambda, v) satisfies A v = lambda v. Non-zero vectors v that meet this relation are eigenvectors of A.
The solver repeatedly factors A - mu I = Q R then updates A ← RQ + mu I. After iterations, the diagonal approaches the eigenvalues.
The accumulated product approximates eigenvectors: V ≈ Q1 Q2 ... Qk. Each eigenpair is validated using ||Av - lambda v||.
Lower residual values mean the computed eigenpair closely satisfies the eigenvalue equation.
How to Use This Calculator
- Paste your matrix as rows on separate lines.
- Separate numbers using commas or spaces.
- Choose iterations, tolerance, and precision.
- Press Solve Eigenvectors to compute results.
- Review eigenvalues, eigenvectors, and residual norms.
- Use the download buttons to export CSV or PDF.
Notes on Accuracy
- Real-only QR iteration may struggle with complex eigenpairs.
- Symmetric matrices often return orthogonal eigenvectors and stable eigenvalues.
- If results look unstable, increase iterations or loosen tolerance.
Matrix input standards and sizing
Enter an n×n matrix using rows separated by new lines and values separated by commas or spaces. The solver supports sizes from 2×2 to 6×6 to keep numerical behavior stable in a lightweight runtime and to ensure readable output tables. Decimals and negative entries are accepted, and consistent row length is required for a valid square matrix.
Iteration controls and convergence signals
Shifted QR iteration updates A by factoring A−μI into Q and R, then forming RQ+μI. Tolerance is 1e−10, while the iteration cap can be raised to 2000. Convergence is tracked by the off‑diagonal Frobenius norm, which shrinks as A approaches triangular form. If that norm stalls, increase iterations.
Residual norms as quality metrics
Each eigenpair is validated with the residual ||Av−λv||. If this value is close to zero, the displayed vector v approximately satisfies the eigenvalue equation. Use this metric to compare solutions across different tolerances, or to detect cases where complex eigenpairs prevent full real convergence. Residuals also reveal when two eigenvalues are nearly equal and vectors become sensitive to small perturbations.
Sorting, normalization, and interpretability
Eigenvectors can be normalized to unit length to simplify comparisons, and eigenvalues can be sorted by value or by absolute magnitude. Sorting by |λ| is often helpful when you care about dominant modes, while ascending order is useful for stability studies and sensitivity checks. When normalization is enabled, sign flips are still possible, because v and −v represent the same direction.
Performance characteristics and practical limits
For an n×n matrix, QR steps are O(n³) per iteration, dominated by matrix products and orthogonalization. With n≤6, the main cost is iteration count, not size, so a tight tolerance can noticeably increase runtime. Symmetric matrices often converge faster and yield near‑orthogonal eigenvectors, whereas poorly scaled matrices may need higher precision and more iterations.
Common workflows in science and engineering
Typical use cases include principal component directions from covariance matrices, vibration mode shapes in structural models, steady‑state analysis of linear recurrences, and ranking problems based on dominant eigenvectors. Exported CSV supports audit trails and downstream plotting, while the PDF snapshot supports reporting and reviews. Pair the chart with the residual column to document both magnitude and accuracy in one view.
FAQs
What matrices work best with this solver?
Real symmetric matrices usually give the cleanest results, with fast convergence and orthogonal eigenvectors. Non‑symmetric matrices may converge more slowly, and matrices with complex eigenvalues may not fully diagonalize in a real-only method.
Why do eigenvectors differ by sign between runs?
Eigenvectors represent directions, so v and −v are equivalent. Small numerical differences or sorting choices can flip signs without changing the meaning. Compare absolute components or use dot products to align vectors consistently.
How should I choose tolerance and iterations?
Start with tolerance 1e-8 to 1e-10 and 200 iterations. If the off-diagonal norm and residuals remain large, raise iterations or loosen tolerance. For tough inputs, increase precision to view small changes reliably.
What does the residual value tell me?
Residual ||Av−λv|| measures how well the displayed pair satisfies the eigenvalue equation. Values near zero indicate a strong solution. If residuals are high, the iteration may not have converged or the matrix may require different scaling.
Why are results missing or unstable for some inputs?
Nearly repeated eigenvalues and poorly scaled entries can make eigenvectors sensitive. Try rescaling the matrix, enabling normalization, and sorting by |lambda|. If the matrix has complex eigenpairs, consider using a complex-capable solver.
What is included in the CSV and PDF exports?
The CSV lists each eigenvalue and its eigenvector components, padded to six columns for consistency. The PDF includes the same values plus iteration count, off-diagonal norm, and residuals, making it suitable for sharing and documentation.