This capstone note closes Topic 9 by tying its entire eigenvalue toolkit back to Stage 3: Theorem 1 of the SVD-existence note proved the SVD exists by reducing it to an eigenvalue problem for — and this note shows that reduction is not merely a proof technique, it is (a refined version of) the actual numerical method: computing an SVD in production software means running exactly the eigenvalue algorithms this topic just built, on a specific reformulation designed to avoid one numerical pitfall the direct approach carries.
Learning objectives. By the end of this note you should be able to:
- explain why computing the SVD via directly is numerically inferior to alternatives;
- state the symmetric eigenvalue reformulation used in practice;
- connect the SVD computation to Hessenberg/tridiagonal reduction and the QR algorithm;
- explain the sensitivity of singular values to perturbations.
In this lesson
- Why not just diagonalise directly?
- The production method
- Sensitivity of singular values
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