Enter Matrix Data
The page uses a single-column flow, while the calculator inputs adapt to 3 columns on large screens, 2 on smaller screens, and 1 on mobile.
Example Data Table
| Case | Matrix | Expected Insight |
|---|---|---|
| Square 3×3 | [3 1 1; -1 3 1; 0 1 2] | Good for checking singular spectrum, rank, and exact reconstruction. |
| Tall 4×3 | [1 0 2; 2 1 3; 0 1 1; 1 1 2] | Useful for reduced representation and least-squares intuition. |
| Wide 3×5 | [1 2 0 1 3; 0 1 1 2 1; 2 0 1 1 0] | Shows nullity, redundancy, and pseudoinverse behavior clearly. |
Formula Used
A = U Σ VT
σi = √λi(ATA)
ui = (A vi) / σi, when σi > 0
Count singular values greater than the selected tolerance.
κ(A) = σmax / σmin,nonzero
Energy(k) = (Σ σi2 from i=1 to k) / (Σ σi2) × 100%
This page computes SVD numerically through symmetric eigendecomposition and then reconstructs orthonormal factors. That lets you inspect stability, compression quality, and pseudoinverse behavior from the same input matrix.
How to Use This Calculator
- Choose the row count, column count, precision, tolerance, and truncation level.
- Paste your matrix into the textarea, one row per line.
- Click Calculate SVD to generate singular values, factor matrices, and diagnostics.
- Review the graph to see spectrum decay and cumulative energy.
- Use the CSV or PDF buttons to export a clean summary.
Frequently Asked Questions
1) What does singular value decomposition show?
It separates a matrix into orthogonal directions and their strengths. That reveals rank, dominant structure, conditioning, redundancy, and how well lower-dimensional approximations can represent the original data.
2) Why are singular values always nonnegative?
They come from square roots of eigenvalues of ATA or AAT. Those matrices are symmetric positive semidefinite, so their eigenvalues cannot be negative.
3) What does the rank tolerance control?
The tolerance decides which very small singular values count as numerical zero. Increasing it lowers the reported rank and can treat noisy directions as negligible.
4) What is the benefit of truncation k?
Top-k truncation keeps only the strongest singular components. This is useful for compression, denoising, latent structure discovery, and reduced models with lower storage cost.
5) Why does the pseudoinverse matter?
The pseudoinverse supports least-squares solutions, minimum-norm estimates, and stable handling of non-square or rank-deficient systems where ordinary inversion fails.
6) What does a large condition number mean?
A large condition number indicates sensitivity. Small perturbations in the matrix or data may produce large changes in solutions, especially in inverse or regression-style problems.
7) Can I use rectangular matrices here?
Yes. The calculator supports rectangular matrices. SVD is especially useful for tall or wide matrices because it still provides orthogonal factors, singular values, and pseudoinverse information.
8) Why might reconstruction error not be exactly zero?
Finite precision arithmetic causes tiny rounding differences. For a well-computed decomposition, the exact reconstruction error should still be very close to zero.