SVD in One Line

• U, V orthogonal; Σ diagonal with σ₁ ≥ σ₂ ≥ … ≥ 0
• Columns of V: input directions
• Columns of U: output directions
• σᵢ: stretch factor along each direction

Why it matters: Every matrix is a rotation, a scaling, and another rotation. Truncating at rank-k gives the best rank-k approximation in Frobenius norm — this is PCA, LoRA, and half of numerical linear algebra.