Reading paths
Every result in this book is proved from results that came earlier, and the build records which. Following those records backwards from a theorem gives the sections a reader actually needs in order to reach it, and nothing else. The paths below are computed that way, from the proofs themselves, each time the book is built.
A path is aimed at a handful of sections, listed on its own page. It contains every section those proofs rest on, and it is closed: nothing on it cites anything off it in a proof. Read it in order and no step is missing.
Each path comes in two lengths. The reading column counts what the proofs need. The with exercises column also counts what the written solutions need, since a solution is the proof of its exercise; it is the longer figure, and the difference is the price of working the problems rather than reading past them.
| Reading path | Reading | With exercises |
|---|---|---|
| Computer science and machine learningThe singular value decomposition and what is built on it: low-rank approximation, the pseudoinverse, principal components, graph Laplacians and PageRank. | 98 | 126+28 |
| Economics and econometricsLeast squares, definiteness and the second-derivative test, linear programming duality and matrix games, Markov chains, and the M-matrices behind an input-output model. | 105 | 134+29 |
| Pure mathematics and algebraCanonical forms over an arbitrary field, quadratic forms and Witt's theorems, the exterior algebra, and the characters of a finite group. | 102 | 117+15 |
| Optimization and operations researchLeast squares, separating hyperplanes, linear programming duality, polytopes and convex functions, the Loewner order, and conjugate gradients. | 107 | 143+36 |
| Physics and quantum mechanicsThe matrix exponential, the functional calculus of a normal operator, Minkowski space, tensor products in quantum mechanics, and the normal modes of a vibrating structure. | 93 | 114+21 |
| Electrical engineering and signal processingOrthogonal expansions, the QR factorization, circulants and the discrete Fourier transform, the singular value decomposition, conditioning, and the fast Fourier transform. | 92 | 102+10 |
| StatisticsLeast squares in theory and in floating point, Gram matrices, two quadratic forms diagonalized at once, and principal components. | 100 | 121+21 |
| Graphics and geometrySigned volume and orientation, the shape of a rotation, the polar decomposition, Euclidean motions, projective transformations, quadrics, and line coordinates in space. | 120 | 143+23 |
| Control theoryThe matrix exponential and linear differential equations, convergent and power-bounded matrices, the Sylvester and Lyapunov equations, the spectral radius, and Krylov subspaces. | 99 | 135+36 |
| Coding theory and cryptographyThe algebra that codes and lattices are built from: changing the field, rank over a finite field, irreducible polynomials, the Chinese remainder theorem, and the Smith normal form. | 65 | 70+5 |
Out of 237 sections that carry a result. None of these lists is a syllabus: it is the smallest set that leaves no proof unfinished. To aim at something else, see the dependency graph, which will compute the same closure for any single section.