Economics and econometrics
Least squares, definiteness and the second-derivative test, linear programming duality and matrix games, Markov chains, and the M-matrices behind an input-output model.
It aims at 11.4 Least Squares and Minimum-Norm Solutions, 14.6 Definiteness and Second Derivatives, 18.6 Linear Programming Duality, 18.7 Matrix Games and the Minimax Theorem, 19.6 Markov Chains, 19.9 M-Matrices.
105 sections to read, of 236. That is everything the proofs of those sections rest on, and nothing else.
133 if you also work the exercises, since a written solution is the proof of its exercise: 28 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
Block Matrices5 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 Chains
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 Matrices
- +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 Matrices
Inner Product Spaces8 of 10
Spectral Theory9 of 11
- 12.1 Schur Triangularization
- 12.2 Self-Adjoint Operators
- 12.3 Normal Operators2 proofs
- 12.4 The Complex Spectral Theorem
- 12.5 The Real Spectral Theorem
- +12.7 Commuting Families
- 12.8 Spectral Resolutions and Functions of a Normal Operator1 proof
- +12.10 The Sylvester Equation
- +12.11 How Far from Normal
Positive Matrices and the Singular Value Decomposition6 of 12
Bilinear and Quadratic Forms6 of 12
Norms and Matrix Analysis6 of 8
Variational Principles and Interlacing4 of 11
Convexity6 of 11
Non-negative Matrices7 of 9
What you can skip
131 of the book's 236 sections are not on this path, and skipping them costs you no proof you need.
Not needed at all: Tensors and Exterior Algebra, Perturbation Theory, Matrix Inequalities, Affine and Projective Geometry, Algebras and Representations, Computation and Applications.
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: 13.12 Projections Revisited.
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.