Linear Algebra — 28.2: Reducing SVD to A^T A
The SVD writes any matrix as A = U Σ V^T, three unknowns in a single equation. This video shows how multiplying A by its transpose cancels one of the orthogonal factors, reducing the entire problem to a symmetric eigenproblem we already know how to solve.
Key concepts covered:
Why A^T A = V Σ² V^T: the U^T U = I cancellation
Columns of V as eigenvectors of A^T A, with singular values as square roots of its eigenvalues
The parallel identity A A^T = U Σ² U^T and why both matrices share the same nonzero eigenvalues
The sign ambiguity trap when computing U and V independently
Recovering U from V using u_i = (1/σ_i) A v_i to lock signs consistently
Why this construction guarantees the SVD exists for every matrix
The historical link between SVD algorithms and symmetric eigensolvers
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SOURCE MATERIALS
The source materials for this video are from • 29. Singular Value Decomposition