SVD: A = UΣVᵀ. Every matrix has an SVD. σᵢ = singular values (always ≥ 0). rank(A) = number of nonzero σᵢ. ||A||₂ = σ₁ (largest). ||A||F = √(Σσᵢ²). Computed via eigendecomposition of AᵀA (eigenvalues = σᵢ²). The most important matrix factorization.
How to Use
Enter a matrix:
- A: Any m×n matrix
- Output: U, Σ, Vᵀ
- Also: Rank, norms, κ
Applications
- PCA (Principal Component Analysis)
- Image compression (low-rank approximation)
- Pseudoinverse: A⁺ = VΣ⁺Uᵀ
- Recommender systems
Low-Rank Approximation
Best rank-k approximation: keep top k singular values, zero the rest. Error = σₖ₊₁. This is the Eckart-Young theorem. Used in image compression and dimensionality reduction.
Step-by-Step Instructions
- 1Select a matrix.
- 2View singular values.
- 3See U and Vᵀ.
- 4Check rank.
- 5Analyze condition.