Overview — Spectral and Factor Methods

A covariance that is not positive semi-definite invents negative variance along some portfolio. Small singular directions in a returns panel can contain noise or useful risk information. A large condition number warns that some input perturbations can greatly change a hedge; it gives a worst-case bound rather than predicting the realised error. This topic owns those three facts: equivalent characterisations of positive (semi-)definiteness and Cholesky, SVD/PCA truncation as optimal factor compression, and condition numbers as the right near-singularity diagnostic for desks.

By the end of this note you should be able to:

• test PSD/PD via eigenvalues and quadratic forms, and explain why a negative eigenvalue of a "covariance" is a data or model failure;
• truncate an SVD / PCA to $k$ factors and quote the Eckart–Young error;
• compute $\kappa(A)$ and read a perturbation bound for regression or a hedge.

1. Contents
2. The big picture