Linear Algebra

Read these first

1. 01Cholesky Decomposition

   Every valid covariance matrix has a 'square root' that turns independent noise into correlated noise, the Cholesky decomposition is how simulations get correlated random numbers out of a stream of uncorrelated ones.

2. 02Projection Matrices and Least Squares

   Fitting a line through data no line can pass through exactly is a shadow problem: the fitted values are literally the shadow of your data on the space of possible fits, cast straight down.

3. 03The Matrix Condition Number

   One number tells you whether solving a system of equations is safe or a numerical minefield: the condition number measures how much a matrix amplifies small errors, and a big one means your answer cannot be trusted.

4. 04The Spectral Theorem for Symmetric Matrices

   Every symmetric matrix, however tangled it looks, is secretly just a stretch along a set of perpendicular directions, the spectral theorem guarantees those directions always exist and are always at right angles.

Then the rest

1. 05Determinants
2. 06Inner Products and Orthogonality
3. 07Linear Independence and Rank
4. 08Vector Spaces and Bases
5. 09Diagonalization and Similarity
6. 10Gram-Schmidt Orthogonalization
7. 11LU Decomposition
8. 12Matrix Calculus and Derivatives
9. 13Matrix Inverses and Linear Systems
10. 14QR Decomposition
11. 15Quadratic Forms
12. 16The Moore-Penrose Pseudoinverse
13. 17Trace and Matrix Norms
14. 18Low-Rank Approximation and the Eckart-Young Theorem
15. 19The Graph Laplacian and Spectral Clustering
16. 20The Marchenko-Pastur Law
17. 21The Matrix Exponential
18. 22The Nearest Correlation Matrix Problem
19. 23The Schur Complement
20. 24The Sherman-Morrison-Woodbury Formula
21. 25Toeplitz and Circulant Matrices