Random Matrix Theory
Spectral norms of random matrices, singular value bounds, Wishart ensembles and the Marchenko–Pastur law.
Level 3 — Graduate
- Spectral Norm of Random Matrices12 min
The net-plus-concentration argument that gives the sqrt(n) + sqrt(p) bound, and the semicircle law for Wigner matrices.
Spectral norm of a matrix with sub-Gaussian entriesBilinear variational form of the spectral normWigner semicircle law - Sample Covariance and the Marchenko–Pastur Law6 min
What the spectrum of an empirical covariance matrix looks like when the dimension is comparable to the sample size, and why that breaks classical statistics.
Marchenko–Pastur lawBulk edges and the conditioning of the sample covarianceThe Marchenko–Pastur density integrates to one
Problems (9)
Open in the browser| Solved | Problem | Type | Difficulty | Time |
|---|---|---|---|---|
| Marchenko–Pastur edges and conditioning | Numerical | 15 min | ||
| Frobenius versus operator norm for random matrices | Numerical | 30 min | ||
| Simulate the Marchenko–Pastur law run | Coding | 1 hour | ||
| Simulate Wigner and Wishart spectra run | Coding | 1 hour | ||
| Catalan numbers are the moments of the semicircle law | Proof | 1 hour | ||
| The Marchenko–Pastur density integrates to one | Proof | 1 hour | ||
| Why ordinary least squares degrades as p approaches n | Proof | 30 min | ||
| The net argument for the spectral norm abyss | Proof | 3 hours | ||
| Derive the Marchenko–Pastur law via the Stieltjes transform abyss | Proof | 1 day |