Σ SVD Calculator

SVD — Singular Value Decomposition

Compute A = UΣVᵀ for any matrix. Shows U (left singular vectors), Σ (singular values), and V (right singular vectors) with rank and condition number.

Σ

Singular Value Decomposition A = UΣVᵀ

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Matrix A
Quick Examples

Singular Value Decomposition

SVD is one of the most important matrix factorizations. Every real m×n matrix A can be written as:

$$A = U\Sigma V^T$$
  • U: m×m orthogonal matrix (left singular vectors)
  • Σ: m×n diagonal matrix (non-negative singular values σ₁ ≥ σ₂ ≥ ... ≥ 0)
  • V: n×n orthogonal matrix (right singular vectors)

Applications

  • Image compression (low-rank approximation)
  • Principal Component Analysis (PCA)
  • Pseudoinverse and least-squares solutions
  • Numerical rank determination
  • Condition number: κ = σ₁/σₙ (ratio of largest to smallest non-zero singular value)