SVD Calculator
Calculate the Singular Value Decomposition (SVD) of a matrix: A = UΣV*
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Singular Value Decomposition (SVD)
Every matrix A can be decomposed into the product of three matrices:
\[ A = U\Sigma V^* \]
Where:
- U is an orthogonal matrix (left singular vectors)
- Σ is a diagonal matrix (singular values)
- V* is the conjugate transpose of V (right singular vectors)
Properties:
- U and V are orthogonal/unitary
- Σ has non-negative diagonal entries
- Singular values are ordered (σ₁ ≥ σ₂ ≥ ...)
- Decomposition is unique up to sign
- Number of non-zero singular values equals rank
Applications:
- Principal Component Analysis (PCA)
- Image compression
- Data dimensionality reduction
- Signal processing
- Machine learning
Geometric Interpretation:
- U represents rotation/reflection
- Σ represents scaling
- V* represents rotation/reflection
- Reveals principal directions