How to Use Singular Value Decomposition Calculator
The Singular Value Decomposition Calculator factors a matrix A = UΣVᵀ and shows each component matrix. Enter the matrix size and values, and the Singular Value Decomposition Calculator computes U, Σ, and Vᵀ, verifies their product reconstructs A, and lists every computational step.
- Set dimensions – Select rows and columns (1–4 each) for matrix A.
- Enter the matrix – Fill each cell with a real number (decimals allowed).
- Read the decomposition – The Singular Value Decomposition Calculator displays U, Σ, and Vᵀ as formatted matrices.
- Check the verification – The tool shows UΣVᵀ so you can confirm it matches A.
- Use singular values – The diagonal entries of Σ are the singular values, sorted largest first.
Formula & Theory - Singular Value Decomposition Calculator
The Singular Value Decomposition Calculator implements the full SVD:
A = U Σ Vᵀ
where:
U ∈ ℝ^(m×m) — left singular vectors (columns are orthonormal)
Σ ∈ ℝ^(m×n) — diagonal matrix of singular values σ₁ ≥ σ₂ ≥ … ≥ 0
Vᵀ ∈ ℝ^(n×n) — right singular vectors (rows are orthonormal)| Symbol | Meaning |
|---|---|
| A | Input m×n real matrix |
| U | Left singular matrix; columns are eigenvectors of AAᵀ |
| Σ | Rectangular diagonal; Σᵢᵢ = σᵢ |
| Vᵀ | Right singular matrix; rows are eigenvectors of AᵀA |
| σᵢ | Singular values: σᵢ = √λᵢ(AᵀA) |
Algorithm inside the Singular Value Decomposition Calculator:
- Compute AᵀA (n×n symmetric matrix).
- Apply the Jacobi eigenvalue algorithm to find all eigenvalues λᵢ and eigenvectors vᵢ of AᵀA. These eigenvectors form the columns of V.
- Sort by descending eigenvalue; singular values σᵢ = √λᵢ.
- Compute the left singular vectors: uᵢ = Avᵢ / σᵢ (for σᵢ > 0).
- Assemble U, Σ, and Vᵀ; compute the verification product UΣVᵀ.
Useful derived quantities:
- Low-rank approximation:
A_k ≈ σ₁u₁v₁ᵀ + … + σₖuₖvₖᵀ - Pseudoinverse:
A⁺ = VΣ⁺Uᵀ(Σ⁺ replaces non-zero σᵢ with 1/σᵢ) - 2-norm:
||A||₂ = σ₁ - Frobenius norm:
||A||F = √(Σσᵢ²)
Assumptions and Limits
The Singular Value Decomposition Calculator supports real matrices up to 4×4. The Jacobi algorithm iterates up to 200 sweeps, which is more than enough for small dense matrices. For singular values near zero (< 10⁻⁹), the corresponding left singular vectors are set to zero vectors. In production, always use a well-tested library (NumPy, LAPACK) for SVD.
Use Cases for Singular Value Decomposition Calculator
The Singular Value Decomposition Calculator is widely used across data science, engineering, and mathematics:
- Dimensionality reduction / PCA – The right singular vectors in Vᵀ are principal directions; singular values indicate variance captured.
- Low-rank matrix approximation – Truncate to the top-k SVD terms for data compression or noise removal.
- Least-squares and pseudoinverse – Compute A⁺ = VΣ⁺Uᵀ to solve Ax = b in the least-squares sense.
- Condition number – κ(A) = σ_max / σ_min measures sensitivity of the linear system.
- Latent Semantic Analysis (LSA) – Apply SVD to a term-document matrix to discover hidden topics.
- Image compression – Represent an image matrix with k dominant singular values/vectors.
The Singular Value Decomposition Calculator gives you all three factor matrices so you can experiment with rank truncation, verify orthogonality of U and V, and directly apply SVD-based formulas.