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Statistical Arbitrage / The Factor-Residual Model and the S-Score

Derivation — Spectral Decomposition and Eigenportfolio Construction

The claim that V=WΛW′V = W\Lambda W' (the spectral decomposition used in eigenportfolio construction) rests on three non-obvious facts: that a symmetric matrix always has real eigenvalues, that its eigenvectors can be chosen orthonormal, and that there is a finite k≤Nk \leq N such that the first kk eigenportfolios capture any desired fraction of total variance. Without these facts the eigenportfolio framework has no foundation — why would eigenvectors, which are defined algebraically, also be variance-maximising portfolios?

This derivation note proves the spectral theorem for symmetric matrices, derives the equivalence of PCA directions with variance maximisers (via Lagrangian optimisation), and constructs the eigenportfolio returns from the eigenvector weights. It is the algebraic supplement to PCA eigenportfolios as statistical risk factors.

In this lesson

    1. The eigenvalue problem
    1. Variance maximisation: the PCA motivation
    1. Eigenportfolio construction algorithm

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