Math Wiki

Understand how geodetic network adjustment led to Cholesky factorization, how its purpose developed, and how to compute it.

Derive conjugate gradients from Hestenes and Stiefel's published system, follow its four steps, and explore residuals, energy error, and preconditioning.

Build perpendicular unit directions by removing projections, preserve successive spans, and understand the difference between classical and modified Gram–Schmidt.

Derive a reflection that zeros a column tail, understand its geometry and stable sign choice, and connect compact reflectors to QR and eigenvalue algorithms.

Factor a matrix into lower and upper triangular matrices to solve linear systems efficiently.

Represent matrices as products of structured factors, and choose between LU, Cholesky, QR, and other decompositions.

Measure matrix amplification with induced norms, distinguish spectral and Frobenius norms, and bound errors in linear systems.

Store orthogonalization as A = QR, solve least squares through a triangular system, and compare Gram–Schmidt, Householder reflections, and Givens rotations.

Read xᵀAx as a scalar quadratic expression, understand its geometry, and connect its curvature to matrix factorizations and optimization.

Eliminate a block of variables and understand the remaining matrix through linear systems, quadratic energy, and Cholesky factorization.

Represent sparse matrices with COO, CSR, and CSC, read their pointer arrays, and choose a format for assembly or computation.

Recognize positive energy in every direction through quadratic forms, eigenvalues, Cholesky factors, and worked examples.

Measure vector size with the 1, 2, and infinity norms, compare their geometry, and choose a meaningful error measure.

Compress a convex combination to at most d + 1 points, starting from Carathéodory’s Fourier-coefficient problem and a constructive weight-elimination proof.

Derive convex sets from mixing production plans, distinguish convex functions from feasible regions, and connect global minima to quadratic curvature.

Prove that linear constraints are impossible by combining them into a contradiction, and connect the certificate to cones and LP duality.

Derive linear models, improving exchanges, and optimality certificates from Kantorovich's machine allocation and Dantzig's assignment problem.

Turn production constraints into optimality certificates, prove LP strong duality with Farkas’ lemma, and derive Lagrange duals and KKT conditions.

Derive basis exchanges, reduced costs, and the ratio test from a historical production problem, then certify the answer and find a feasible start.