Eigen Decomposition Calculator

Example Data Table

Try this symmetric 3×3 matrix to verify output and reconstruction error.

Formula Used

How to Use This Calculator

1. Select the matrix size and enter A values.
2. Keep “Symmetric solver” enabled for full decomposition.
3. Set tolerance and maximum iterations as needed.
4. Click compute to view eigenvalues and eigenvectors.
5. Download CSV or PDF for reporting and documentation.

Numerical Stability and Symmetry

Eigen decomposition is most reliable when the matrix is real and symmetric, because eigenvalues are guaranteed real and eigenvectors can be chosen orthonormal. This calculator offers a symmetric solver based on Jacobi rotations, which progressively removes off‑diagonal energy. For engineering matrices, symmetry often comes from energy formulations where A equals Aᵀ by construction; verifying symmetry avoids misleading complex modes. If your data is “almost symmetric” from measurement noise, the optional symmetrization step uses (A + Aᵀ)/2 to reduce drift and improve convergence.

Choosing Tolerance and Iterations

Tolerance controls when the algorithm stops: in symmetric mode it monitors the largest absolute off‑diagonal entry, and in general mode it monitors the off‑diagonal norm after QR steps. A tighter tolerance can sharpen results but may require more iterations. For 2×2 to 6×6 matrices, values like 1e‑10 to 1e‑12 are practical for double‑precision arithmetic. If convergence is not reached, increase the iteration limit or relax tolerance slightly.

Interpreting Eigenvalues in Practice

Eigenvalues summarize how the matrix scales directions, which is why they appear in vibration modes, covariance analysis, and linear transformations. Positive eigenvalues often indicate energy, variance, or stiffness, while negative eigenvalues can signal instability or saddle behavior. When the matrix represents covariance, very small eigenvalues imply near‑redundant features. Sorting helps you read dominant modes first, especially when comparing scenarios across datasets.

Understanding Eigenvectors and Orthonormality

In symmetric mode, eigenvectors are returned as columns of V and should satisfy VᵀV ≈ I. The orthogonality diagnostic reports ||VᵀV − I||F, highlighting loss of orthonormality caused by finite precision or early stopping. Each eigenvector direction is only defined up to sign, so v and −v are equivalent. For repeated or nearly repeated eigenvalues, eigenvectors may rotate within the corresponding subspace without changing correctness.

Reporting and Validation Workflow

A practical validation is reconstruction: Â = VDVᵀ should closely match the input matrix when symmetric mode is applicable. The calculator reports ||A − Â||F to quantify this match and makes it easy to export results for peer review. Use the CSV download for audit trails and the PDF export for reports. When documenting outcomes, record matrix size, tolerance, iteration count, and the sorting order.

FAQs

1) What matrices does this calculator handle best?

It performs best on real symmetric matrices, where it returns real eigenvalues and orthonormal eigenvectors. For general matrices, it reports real eigenvalue estimates and focuses on stability and diagnostics rather than full eigenvectors.

2) Why can eigenvectors look different after re-running?

Eigenvectors can flip sign without changing meaning, so v and −v are equivalent. With repeated or nearly repeated eigenvalues, any orthonormal basis of the eigenspace is valid, so vectors may rotate within that subspace.

3) What does the reconstruction error measure?

In symmetric mode it computes ||A − VDVᵀ||F. A small value indicates the returned eigenpairs reproduce the original matrix closely. Larger values suggest early stopping, insufficient iterations, or data that is not truly symmetric.

4) When should I enable “Enforce symmetry”?

Enable it when A should be symmetric but rounding, measurement noise, or manual entry causes small asymmetries. The calculator replaces A with (A + Aᵀ)/2, often improving convergence and making outputs easier to interpret.

5) Why are eigenvectors not shown in general mode?

For non-symmetric matrices, eigenvectors may be complex and can be sensitive to numerical noise. To keep the tool reliable in a browser-friendly implementation, general mode reports real eigenvalue estimates and convergence diagnostics only.

6) How should I choose tolerance for typical work?

For quick checks, try 1e‑8 to 1e‑10. For tighter verification on small matrices, use 1e‑10 to 1e‑12 and raise the iteration limit if needed. Always interpret tolerance together with reconstruction and orthogonality diagnostics.