Linear Algebra — 28.3: Computing the SVD: Two Worked Examples
Two complete SVD computations side by side: a full-rank 2x2 matrix and a rank-one matrix where a singular value vanishes. Along the way we see why sign choices in U are not free, how zero singular values expose the null space, and why the factorization never breaks on singular matrices.
Key concepts covered:
The SVD recipe: AᵀA gives V and σᵢ², then A vᵢ = σᵢ uᵢ gives U
Worked example: A = [[4,4],[-3,3]] with singular values √32 and √18
The sign trap: why u_i must be computed from A v_i, not guessed from AAᵀ
Worked example: rank-one matrix [[4,3],[8,6]] with a zero singular value
Identifying the four fundamental subspaces by inspection
How a zero in Σ annihilates a column of U and a row of Vᵀ
Reconstructing a rank-one matrix from a single outer product σ₁ u₁ v₁ᵀ
The number of zero singular values equals the dimension of the null space
Why zero and small singular values are the foundation of low-rank approximation, PCA, and the pseudoinverse
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SOURCE MATERIALS
The source materials for this video are from • 29. Singular Value Decomposition