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SVD Visualizer

Singular Value Decomposition — A = U Σ Vᵀ

Linear Algebra Matrix Decomposition Data Science Math

Preset:

A = [

]

σ₁ = 1.000 σ₂ = 1.000 det(A) = 1.000 rank = 2 ‖A‖₂ = 1.000 cond = 1.000

🔢 Singular Value Decomposition

Every real matrix A can be decomposed as: A = U Σ Vᵀ

U (m×m) — left singular vectors (orthogonal columns, rotation/reflection) · Σ (m×n) — diagonal matrix of singular values σ₁ ≥ σ₂ ≥ 0 · Vᵀ (n×n) — right singular vectors (orthogonal rows, rotation/reflection).

Geometric interpretation: A maps the unit circle to an ellipse with semi-axes σ₁ (along u₁) and σ₂ (along u₂). The columns of V are the input directions that get the largest scaling. For det > 0 the transformation preserves orientation; det < 0 flips it.

Applications: principal component analysis (PCA), image compression, pseudo-inverse, noise reduction, latent semantic analysis. The nuclear norm ‖A‖* = Σ σᵢ is the trace-norm used in low-rank regularization.