LA

Linear algebra

Projection, eigenstructure and decompositions — the machinery behind OLS, PCA and risk models.

1
Vectors and matrices
Rank, trace, determinant and the special matrix types.
- 1.1Rank, trace and determinant: what each one measures11 min
2
Linear systems and spaces
The four fundamental subspaces and rank–nullity.
- 2.1The four fundamental subspaces, and when Ax = b has a solution11 min
3
Orthogonality and projection
Gram–Schmidt, QR, and OLS derived as a projection.
- 3.1Orthogonality, QR and OLS as a projection12 min
4
Eigenvalues and eigenvectors
Diagonalisation, the spectral theorem and power iteration.
- 4.1Eigenvalues, diagonalisation and the spectral theorem12 min
5
Definiteness and covariance
Why covariance matrices are PSD, and generating correlated normals.
- 5.1Definiteness, Cholesky and generating correlated normals12 min
6
SVD and dimensionality reduction
Low-rank approximation, PCA derived two ways, and factor models.
- 6.1PCA, covariance matrices and what an eigenvalue is telling you14 min
7
Matrix calculus
The gradient identities behind OLS, ridge and portfolio solutions.
- 7.1Matrix calculus: the identities behind OLS, ridge and portfolios12 min