Optimization and operations research
Least squares, separating hyperplanes, linear programming duality, polytopes and convex functions, the Loewner order, and conjugate gradients.
It aims at 11.4 Least Squares and Minimum-Norm Solutions, 13.5 The Loewner Order, 18.6 Linear Programming Duality, 18.9 Polytopes, 18.11 Three Classical Inequalities, 24.7 Steepest Descent and Conjugate Gradients.
107 sections to read, of 237. That is everything the proofs of those sections rest on, and nothing else.
143 if you also work the exercises, since a written solution is the proof of its exercise: 36 more sections.
What to read
In reading order. Bold marks a section the path is aimed at; + marks one that only the exercises need. The note beside a section says when the whole path hangs on a single proof citing it.
Preliminaries10 of 10
Vector Spaces9 of 9
Linear Maps5 of 5
Linear Systems and Matrices7 of 7
Matrices of Maps5 of 5
Duality5 of 6
Polynomials7 of 7
Determinants8 of 10
- 7.2 Multilinear Alternating Forms
- 7.3 Permutations and Sign
- 7.4 Existence and Uniqueness of the Determinant
- 7.5 Row Operations and Multiplicativity
- 7.6 Cofactor Expansion, the Adjugate, and Cramer's Rule
- 7.7 Minors, Rank, and Cauchy–Binet1 proof
- 7.8 Special Determinants
- 7.10 The Characteristic Polynomial
Block Matrices4 of 5
Eigenvalues and Diagonalization11 of 11
- 9.1 Invariant Subspaces
- 9.2 Eigenvalues and Eigenvectors
- 9.3 Algebraic and Geometric Multiplicity
- 9.4 Diagonalization
- 9.5 The Minimal Polynomial
- 9.6 The Cayley–Hamilton Theorem
- 9.7 Triangularization
- 9.8 Primary Decomposition
- +9.9 Commuting Operators
- 9.10 Spectral Mapping, Products in Both Orders, Left Eigenvectors
- 9.11 Powers, Recurrences, and Markov Chains1 proof
Canonical Forms9 of 10
- 10.1 Generalized Eigenspaces
- 10.2 Nilpotent Operators2 proofs
- 10.3 The Jordan Canonical Form
- +10.4 The Real Jordan Form and the Weyr Form
- 10.5 Cyclic Subspaces and Companion Matrices2 proofs
- +10.6 The Rational Canonical Form
- +10.7 Polynomial Matrices and the Smith Normal Form
- 10.9 Functions of Matrices, the Exponential, and Linear Differential Equations2 proofs
- 10.10 Convergent and Power-Bounded Matrices1 proof
Inner Product Spaces9 of 10
- 11.1 Inner Products and Norms
- 11.2 Orthonormal Bases and Gram–Schmidt
- 11.3 Orthogonal Complements and Projections
- 11.4 Least Squares and Minimum-Norm Solutions
- 11.5 The Riesz Representation Theorem2 proofs
- 11.6 The Adjoint
- 11.7 Isometries and Unitary Maps
- +11.8 QR, Householder, and Hessenberg
- 11.9 Bessel, Parseval, and Fourier Series1 proof
Spectral Theory8 of 11
Positive Matrices and the Singular Value Decomposition6 of 13
Bilinear and Quadratic Forms5 of 12
Norms and Matrix Analysis7 of 8
Variational Principles and Interlacing8 of 11
Convexity10 of 11
- 18.1 Convex Sets and Hulls
- 18.2 Carathéodory's Theorem1 proof
- 18.3 Separating Hyperplanes
- +18.4 Gauges, Support Functions, and Dual Norms
- 18.5 Cones and Farkas's Lemma1 proof
- 18.6 Linear Programming Duality
- 18.8 Extreme Points2 proofs
- 18.9 Polytopes
- 18.10 Convex Functions2 proofs
- 18.11 Three Classical Inequalities
Nonnegative Matrices5 of 9
Perturbation Theory1 of 11
Computation and Applications4 of 13
What you can skip
130 of the book's 237 sections are not on this path, and skipping them costs you no proof you need.
Not needed at all: Tensors and Exterior Algebra, Matrix Inequalities, Affine and Projective Geometry, Algebras and Representations.
Cited in passing by sections on the path — in a statement, a remark or an example, never in a proof — so they are context and not prerequisites: 11.10 Orthogonal Polynomials, 13.13 Projections Revisited, 20.4 The Bauer–Fike Theorem, 24.1 Floating Point and Backward Error.
Computed from the book's proofs when the book was built, and checked: every result a proof on this path uses is proved by another section on this path. The other paths are listed on the reading paths page.