Computer science and machine learning
The singular value decomposition and what is built on it: low-rank approximation, the pseudoinverse, principal components, graph Laplacians and PageRank.
It aims at 13.11 Low-Rank Approximation, 13.12 The Moore–Penrose Pseudoinverse, 19.8 PageRank, 24.10 Principal Components, 24.11 Graph Laplacians.
98 sections to read, of 237. That is everything the proofs of those sections rest on, and nothing else.
126 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
Duality2 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 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 Theorem1 proof
- 11.6 The Adjoint
- 11.7 Isometries and Unitary Maps
- +11.8 QR, Householder, and Hessenberg
- 11.9 Bessel, Parseval, and Fourier Series2 proofs
Spectral Theory10 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 Operator
- +12.9 Circulants and the Discrete Fourier Transform
- +12.10 The Sylvester Equation
- +12.11 How Far from Normal
Positive Matrices and the Singular Value Decomposition8 of 13
- 13.1 Positive Semidefinite and Positive Definite Matrices
- 13.2 Square Roots and the Cholesky Factorization1 proof
- +13.4 Two Forms at Once
- 13.5 The Loewner Order1 proof
- +13.6 Positive Matrices in Blocks
- 13.9 The Singular Value Decomposition
- 13.11 Low-Rank Approximation
- 13.12 The Moore–Penrose Pseudoinverse
Norms and Matrix Analysis8 of 8
Variational Principles and Interlacing6 of 11
Convexity1 of 11
- 18.1 Convex Sets and Hulls2 proofs
Nonnegative Matrices2 of 9
Computation and Applications4 of 13
What you can skip
139 of the book's 237 sections are not on this path, and skipping them costs you no proof you need.
Not needed at all: Bilinear and Quadratic Forms, Tensors and Exterior Algebra, Perturbation Theory, 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: 5.4 The Dual Map and the Transpose, 13.13 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.