Pure mathematics and algebra
Canonical forms over an arbitrary field, quadratic forms and Witt's theorems, the exterior algebra, and the characters of a finite group.
It aims at 10.6 The Rational Canonical Form, 10.7 Polynomial Matrices and the Smith Normal Form, 14.8 Witt's Extension and Cancellation Theorems, 15.10 The Exterior Algebra, and Clifford at Last, 23.8 Characters and Orthogonality.
102 sections to read, of 236. That is everything the proofs of those sections rest on, and nothing else.
117 if you also work the exercises, since a written solution is the proof of its exercise: 15 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
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–Binet2 proofs
- 7.8 Special Determinants
- 7.10 The Characteristic Polynomial
Block Matrices5 of 5
Eigenvalues and Diagonalization9 of 11
Canonical Forms8 of 10
- 10.1 Generalized Eigenspaces
- 10.2 Nilpotent Operators2 proofs
- 10.3 The Jordan Canonical Form1 proof
- +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 Equations
Inner Product Spaces7 of 10
Spectral Theory7 of 11
Positive Matrices and the Singular Value Decomposition1 of 12
Bilinear and Quadratic Forms8 of 12
Tensors and Exterior Algebra7 of 11
Algebras and Representations8 of 10
What you can skip
134 of the book's 236 sections are not on this path, and skipping them costs you no proof you need.
Not needed at all: Norms and Matrix Analysis, Variational Principles and Interlacing, Convexity, Non-negative Matrices, Perturbation Theory, Matrix Inequalities, Affine and Projective Geometry, 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: 14.11 Minkowski Space, 15.7 Symmetric Powers.
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.