Eigenvalue
An eigenvalue λ of A satisfies A v = λ v. In risk, eigenvalues of the covariance matrix are the variances of the principal components — they tell you how many true factors you have.
Definition
The spectrum of a covariance is nonnegative. A large condition number λ_max/λ_min means inversion is fragile. Random-matrix theory (Marčenko–Pastur) gives a noise bulk so you can keep only eigenvalues above the edge.
Why it matters
Cleaning Σ is an eigenvalue problem. So is stability of a linear dynamical system (rates of decay).
Case
A 400×400 sample covariance from 80 days has many near-zero eigenvalues that are estimation junk. Using them in an inverse explodes weights.
How to read it
Plot the scree against the MP bound. If your ‘signal’ sits in the bulk, it is not signal.
Key papers
Landmark references with real publisher or arXiv links — not token-overlap noise.
- Distribution of Eigenvalues for Some Sets of Random MatricesMarčenko, Pastur · 1967 · Math. USSR-SbornikThe noise bulk for sample covariances.