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Linear Algebra, Optimisation and Numerical Methods / Low-Rank Approximation, PCA, and Applications

The Eckart-Young Theorem: Best Rank-k Approximation

Factor compression replaces a centred returns panel by a lower-rank matrix. The SVD gives a candidate; the Eckart–Young theorem proves it minimises reconstruction error at the chosen rank. This is an in-sample matrix-approximation result. It does not prove that the retained factors forecast returns or capture every financially important hedge.
Learning objectives. By the end of this note you should be able to:
  • construct the rank-kk truncated SVD AkA_k;
  • state the Eckart-Young theorem in both Frobenius and spectral norm;
  • compute the approximation error directly from the discarded singular values;
  • decide how many terms kk to keep for a target approximation quality.

In this lesson

  1. The truncated SVD

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