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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 ━━━━━━━━━━━━━━━━━━━━━━━━ SOURCE MATERIALS The source materials for this video are from https://www.youtube.com/watch?v=TX_vooSnhm8 Attribution: MIT OpenCourseWare, 18.06 Linear Algebra, Spring 2010, Prof. Gilbert Strang — https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/ — License: CC BY-NC-SA 4.0. Adapted (rewritten, narrated and animated) by Ludium; changes were made. License: This video is licensed by Ludium under CC BY-NC-SA 4.0 (https://creativecommons.org/licenses/by-nc-sa/4.0/).

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