SVD & Eigenvalues

Singular Value Decomposition

Any matrix A factors as:

U = left singular vectors, Σ = singular values (σ₁ ≥ σ₂ ≥ 0), V = right singular vectors. The unit circle maps to an ellipse with semi-axis lengths σ₁, σ₂.

Eigenvalues

For square A, eigenvalues satisfy Av = λv. For 2×2: λ² − tr(A)λ + det(A) = 0. Singular values σᵢ = √λᵢ(AᵀA).

3D SVD

For A ∈ ℝ³ˣ³: A = UΣVᵀ where Σ = diag(σ₁,σ₂,σ₃). The unit sphere maps to an ellipsoid with semi-axis lengths σᵢ.

Symmetric matrices have real eigenvalues (Spectral Theorem). General matrices may have complex eigenvalues indicating rotation.