Graphics and geometry
Signed volume and orientation, the shape of a rotation, the polar decomposition, Euclidean motions, projective transformations, quadrics, and line coordinates in space.
It aims at 7.9 Orientation and Volume, 12.6 The Shape of an Orthogonal Matrix, 13.10 Polar Decomposition and Takagi's Factorization, 22.4 Euclidean Motions, 22.5 Affine and Euclidean Quadrics, 22.7 Projective Transformations, 22.10 Conics and Lines in Space.
120 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: 23 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
Duality6 of 6
Polynomials7 of 7
Determinants9 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–Binet
- 7.8 Special Determinants
- 7.9 Orientation and Volume
- 7.10 The Characteristic Polynomial
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 Operators2 proofs
- 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 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 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 Theorem1 proof
- 11.6 The Adjoint
- 11.7 Isometries and Unitary Maps
- 11.8 QR, Householder, and Hessenberg
- +11.9 Bessel, Parseval, and Fourier Series
Spectral Theory9 of 11
- 12.1 Schur Triangularization
- 12.2 Self-Adjoint Operators
- 12.3 Normal Operators
- 12.4 The Complex Spectral Theorem
- 12.5 The Real Spectral Theorem
- 12.6 The Shape of an Orthogonal Matrix
- 12.7 Commuting Families1 proof
- 12.8 Spectral Resolutions and Functions of a Normal Operator
- +12.11 How Far from Normal
Positive Matrices and the Singular Value Decomposition7 of 13
Bilinear and Quadratic Forms7 of 12
Tensors and Exterior Algebra5 of 11
Norms and Matrix Analysis7 of 8
Variational Principles and Interlacing7 of 11
Convexity2 of 11
Affine and Projective Geometry7 of 10
What you can skip
117 of the book's 237 sections are not on this path, and skipping them costs you no proof you need.
Not needed at all: Nonnegative Matrices, Perturbation Theory, Matrix Inequalities, 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: 13.13 Projections Revisited, 15.7 Symmetric Powers, 15.9 The Determinant as a Top Exterior Power, 15.10 The Exterior Algebra, and Clifford at Last, 22.8 Duality.
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.