⚡ SVD Calculator
Compute Singular Value Decomposition A = UΣVᵀ
How to Use This Calculator
Enter Matrix
Input your matrix (works best for 2×2).
Calculate
Get singular values and Σ matrix.
Formula
A = UΣVᵀ
Orthogonal × Diagonal × Orthogonal
Definition:
SVD factors any matrix A into three matrices: U (orthogonal), Σ (diagonal with singular values), and Vᵀ (orthogonal transpose).
Properties:
- U: columns are left singular vectors (eigenvectors of AAᵀ)
- Σ: diagonal matrix with singular values σ₁ ≥ σ₂ ≥ ... ≥ 0
- V: columns are right singular vectors (eigenvectors of AᵀA)
- Works for any m×n matrix
About SVD Calculator
Singular Value Decomposition (SVD) is one of the most important matrix factorizations. It decomposes any matrix into three matrices revealing structure, rank, and geometric properties. SVD is widely used in data compression, least squares, principal component analysis, and more.
When to Use SVD
- PCA: Principal component analysis
- Data Compression: Low-rank approximations
- Least Squares: Pseudoinverse computation
- Rank Reduction: Noise filtering
- Image Processing: Compression and denoising
Why Use Our Calculator?
- ✅ Simplified: Computes singular values for 2×2 matrices
- ✅ Educational: Helps understand SVD
- ✅ Visual: Shows singular values clearly
- ✅ Free: No registration required
Key Concepts
- Singular Values: Non-negative square roots of eigenvalues of AᵀA
- Rank: Number of nonzero singular values
- Orthogonal: U and V are orthogonal/unitary matrices
- Diagonal: Σ contains singular values in descending order
- Uniqueness: SVD is essentially unique (up to column signs)
Applications
Dimension Reduction: Use top k singular values/vectors for approximation.
Low-Rank Approximation: A ≈ UₖΣₖVₖᵀ using largest singular values.
Frequently Asked Questions
What is SVD?
Singular Value Decomposition factors any m×n matrix A into A = UΣVᵀ where U and V are orthogonal and Σ is diagonal with non-negative singular values.
How is SVD different from eigendecomposition?
Eigendecomposition requires square matrices and eigenvectors. SVD works for any matrix size and decomposes into singular values/vectors. Singular values are always real and non-negative.
What are singular values?
Singular values σᵢ are square roots of eigenvalues of AᵀA. They measure transformation scaling along principal axes and are always non-negative real numbers.
How is SVD related to rank?
Rank equals the number of nonzero singular values. SVD provides an optimal low-rank approximation by keeping only the largest singular values.
Can SVD be used for data compression?
Yes! By keeping only the top k singular values (and corresponding vectors), you get a rank-k approximation that compresses data while preserving the most important information.