linear algebra

Explains the Neumann series, its convergence condition, and how it approximates matrix inverses like (I - aA)^{-1}, with a Python implementation example.

Derives the spectral decomposition of symmetric matrices from eigenvalues and eigenvectors, relates it to SVD, and gives a Python implementation example.

Compiles useful matrix formulas from the Matrix Cookbook covering transpose, trace, determinant, differentiation, and the Woodbury inversion identity.

Explains matrix cofactors and cofactor expansion with worked examples, and implements determinant calculation via cofactors in Python.

Explains five matrix decompositions—Cholesky, LU, QR, spectral, and singular value decomposition—with theory and Python implementation examples.

Explains properties of the Rayleigh quotient for symmetric matrices, proves its max/min via Lagrange multipliers, with a Python example.

Explains SVD and PCA fundamentals with NumPy code examples, and how SVD enables low-rank approximation for recommender system rating matrices.

Explains the definition and computation of the pseudo-inverse for matrices that lack a true inverse due to rank deficiency, with a Python example.

Explains singular value decomposition with a recommender system's user-item matrix, and how low-rank approximation predicts user preferences.