U: Unitary matrix (columns are left singular vectors)
Σ: Diagonal matrix with non-negative singular values
V*: Conjugate transpose of V (right singular vectors)
σ₁ ≥ σ₂ ≥ ... ≥ σᵣ ≥ 0
Singular Value Decomposition (SVD) is a factorization of any matrix into three matrices. It is widely used in data analysis, image processing, and machine learning.
⚠SVD always exists for any matrix and is numerically stable. The number of non-zero singular values equals the rank of the matrix.
What is SVD Decomposition?
Singular Value Decomposition (SVD) factorizes any matrix A into A = UΣV*, where U and V are unitary matrices and Σ is a diagonal matrix with non-negative singular values. SVD reveals important structural information about the matrix.
Always Exists
SVD exists for any m×n matrix, even rectangular or singular.
Numerically Stable
SVD computation is numerically stable and reliable.
Rank Information
Number of non-zero singular values = rank(A).
Unitary Matrices
U and V are orthogonal/unitary, preserving inner products.
Example: A = [[1,2],[3,4]]
U: [[-0.4046,-0.9145],[-0.9145,0.4046]]
Σ: [[5.4649,0],[0,0.3659]]
V*: [[-0.5760,-0.8174],[-0.8174,0.5760]]
Verify: U × Σ × V* = A ✓
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