PCA Without the Fundamental Theorem of Algebra
Proves the real spectral theorem using compactness and Lagrange multipliers, then develops SVD, optimal low-rank approximation, and PCA—without invoking the Fundamental Theorem of Algebra.
The notebook
Extended lecture notes.
I sometimes expand lecture notes into a more systematic treatment, filling in proofs and connecting ideas. This is a selection of the notes I’ve extended.
Mathematics of machine learning
Gradient Descent: What Converges, and How Fast?
A proof-based account of standard convergence guarantees: smooth nonconvex objectives, convexity, strong convexity, and the effect of stochastic gradient noise.
Spectral Clustering Through Courant–Fischer
Develops k-means and spectral clustering from an optimization viewpoint, with a proof of the symmetric-matrix Courant–Fischer theorem and a random-walk interpretation.
Ridge Regression and the Influence of a Single Point
Derives the Lagrange multiplier condition geometrically, connects constrained and penalized ridge regression, and uses Sherman–Morrison to derive Cook’s distance. Ends with autoregression.
Probability and stochastic processes
From Martingales to Diffusion Generators
Builds from conditional expectation to stochastic integration and diffusion generators, then connects scale functions, Green kernels, and boundary-value problems.
Mathematical finance
Black–Scholes Through Replication
Derives European call pricing through risk-neutral valuation and replication, then examines delta, early exercise, and a perpetual American put.
These are personal notes, not official course materials. Sources are acknowledged in each piece. Corrections are welcome.