How to Use Singular Value Calculator
The Singular Value Calculator extracts all singular values from a matrix in a few clicks. Set the matrix dimensions, enter the values, and the Singular Value Calculator displays every singular value sorted from largest to smallest, along with the intermediate computation steps.
- Set dimensions – Choose rows and columns (1–5 each).
- Enter matrix A – Fill each cell with a real number.
- Read the singular values – The Singular Value Calculator lists σ₁ ≥ σ₂ ≥ … ≥ σₖ, where k = min(m,n).
- Check the spectral norm – The largest singular value σ₁ equals the 2-norm of the matrix.
- Review the steps – Trace the computation from A through AᵀA to eigenvalues to singular values.
Formula & Theory - Singular Value Calculator
The Singular Value Calculator uses the eigenvalue approach to singular values:
σᵢ = √λᵢ(AᵀA), λᵢ ≥ 0
where λ₁ ≥ λ₂ ≥ … are eigenvalues of AᵀA (or AAᵀ for m < n).| Symbol | Meaning |
|---|---|
| A | Input m×n real matrix |
| Aᵀ | Transpose of A |
| AᵀA | n×n symmetric positive semi-definite Gram matrix |
| λᵢ | Eigenvalues of AᵀA |
| σᵢ | Singular values of A (σᵢ = √λᵢ) |
| k | min(m,n); number of singular values |
Why AᵀA? The matrix AᵀA is always symmetric and positive semi-definite, so all its eigenvalues are real and non-negative. Their square roots are the singular values. The Singular Value Calculator uses the Jacobi QR algorithm with Wilkinson shift to find these eigenvalues efficiently and stably.
Key relationships:
||A||₂ = σ₁(spectral/2-norm equals largest singular value)||A||F = √(σ₁² + σ₂² + … + σₖ²)(Frobenius norm)rank(A) = number of non-zero singular valuesκ(A) = σ₁ / σₖ(condition number for square A)
Assumptions and Limits
Eigenvalues smaller than −10⁻⁹ (which can arise from rounding) are clamped to zero before taking the square root. The Singular Value Calculator supports matrices up to 5×5. For larger matrices, use a numerical library.
Use Cases for Singular Value Calculator
The Singular Value Calculator is valuable in many disciplines:
- Condition number estimation – Divide the largest singular value by the smallest to measure numerical sensitivity.
- Matrix rank – Count the non-zero singular values to determine the rank.
- PCA preprocessing – Singular values are the square roots of principal component variances; the Singular Value Calculator shows their relative magnitudes.
- Data compression – In low-rank approximation, keeping only the largest k singular values controls the approximation quality.
- Control systems – The Hankel singular values determine the order of a reduced model.
- Machine learning – Check for near-zero singular values to detect collinearity in feature matrices.
After using the Singular Value Calculator to get the singular values, you can assess how many are significant (threshold them by relative magnitude) and use them to bound error in any computation involving the matrix.