Singular Value Decomposition (SVD) Calculator
Compute the SVD of a matrix: A = U Σ V^T. U has orthonormal columns, Σ contains the singular values (non-negative), and V^T has orthonormal rows.
Notes
What is SVD?
Every real m×n matrix A has a Singular Value Decomposition A = UΣV^T. U (m×k) has orthonormal columns (left singular vectors), Σ (k×k) is diagonal with non-negative singular values σ₁ ≥ σ₂ ≥ ... ≥ 0, and V^T (k×n) has orthonormal rows (right singular vectors).
| Factor | Dimensions | Meaning |
|---|---|---|
| U | m × k | Left singular vectors (orthonormal columns) |
| Σ | k × k | Diagonal: singular values σ₁ ≥ ... ≥ σₖ ≥ 0 |
| V^T | k × n | Right singular vectors (orthonormal rows) |
Key fact: the number of nonzero singular values equals rank(A). The largest singular value equals the matrix 2-norm ‖A‖₂.
- Matrix Calculations Guide — SVD — In-depth notes on SVD with worked examples
Frequently Asked Questions
What are singular values?
Singular values σᵢ are the square roots of the eigenvalues of A^T A (or AA^T). They are always non-negative real numbers.
What is SVD used for?
Principal Component Analysis (PCA), image compression, least-squares solutions, and computing matrix rank (number of nonzero singular values).
How do singular values relate to eigenvalues?
For a symmetric positive definite matrix, singular values equal eigenvalues. In general they differ: singular values of A are square roots of eigenvalues of A^T A, not eigenvalues of A itself.