Singular Value Decomposition

Factor a matrix as U Σ Vᵀ

About this calculator

The Singular Value Decomposition Calculator factors any m×n matrix (up to 6×6) into A = U Σ Vᵀ, computing the thin SVD via the symmetric eigendecomposition of AᵀA. It lists the singular values in descending order together with the orthonormal U and Vᵀ factors, and derives the numeric rank, condition number, Frobenius norm, and spectral norm. Rank-deficient matrices are handled gracefully — zero singular values are snapped exactly to 0 and the condition number is reported as ∞.

How to use the Singular Value Decomposition calculator

  1. Enter the values for your problem into the input fields.
  2. Read the result — it updates instantly as you type.
  3. Check the formula and the visual explanation to follow how the answer was found.
  4. Copy the page URL to share the exact calculation.

Common examples

  • Diagonal [[3,0],[0,2]] → σ₁ = 3, σ₂ = 2, condition number 1.5, Frobenius norm ≈ 3.605551
  • Shear [[1,1],[0,1]] → σ₁ ≈ 1.618034 and σ₂ ≈ 0.618034 (the golden ratio and its reciprocal)
  • 2×3 matrix [[1,0,1],[0,1,1]] → thin SVD with k = 2: σ = 1.732051 (√3) and 1
  • Rank-1 [[1,2],[2,4]] → σ₁ = 5, σ₂ = 0, rank 1, condition number ∞

Frequently asked questions

How does the Singular Value Decomposition calculator work?

Enter your values and the calculator applies the exact mathematical method for this problem, showing the result together with the formula it used. Everything is computed with high-precision arithmetic, so the answer you see is not limited by ordinary floating-point rounding.

When would I use the Singular Value Decomposition calculator?

It is useful for homework and exam preparation, for checking work done by hand, and whenever a step in a larger problem needs this calculation done quickly and reliably.

How accurate are the results?

Calculations run at 30 significant digits of precision internally. Displayed values are rounded for readability, but the underlying result is far more precise than a typical hand or pocket-calculator computation.

Can I share a specific calculation with someone else?

Yes. Every value you enter updates the URL, so copying the address bar and sharing the link reopens the exact same calculation for whoever clicks it.