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

Principal Component Analysis: the Full Eigenportfolio Derivation

LA-I's PCA-first-look note established that PCA is the SVD applied to a mean-centred data matrix, and explicitly deferred three things to this unit: the full eigenportfolio derivation, choosing K≪NK\ll N components rigorously, and closing the existing PCA pillar-support stub (staged in topics-queue.md) needed by the portfolio-management unit's APT statistical-factors topic. This note delivers all three, now that the Eckart-Young theorem supplies the missing optimality guarantee LA-I's preview could only assert informally.
Learning objectives. By the end of this note you should be able to:
  • prove the first principal component maximises variance among unit-length portfolios;
  • prove successive principal components maximise variance subject to orthogonality;
  • state precisely why choosing the top KK components is the Eckart-Young-optimal choice;
  • construct and interpret an eigenportfolio.

In this lesson

  1. Variance maximisation, proved
  2. Choosing K≪NK\ll N
  3. Eigenportfolios

Before you start

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