Physics and quantum mechanics
The matrix exponential, the functional calculus of a normal operator, Minkowski space, tensor products in quantum mechanics, and the normal modes of a vibrating structure.
It aims at 10.9 Functions of Matrices, the Exponential, and Linear Differential Equations, 12.8 Spectral Resolutions and Functions of a Normal Operator, 14.11 Minkowski Space, 15.11 Tensors in Quantum Mechanics, 24.9 Vibrations.
93 sections to read, of 237. That is everything the proofs of those sections rest on, and nothing else.
114 if you also work the exercises, since a written solution is the proof of its exercise: 21 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 Diagonalization9 of 11
Canonical Forms8 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 Equations
Inner Product Spaces8 of 10
Spectral Theory8 of 11
Positive Matrices and the Singular Value Decomposition6 of 13
Bilinear and Quadratic Forms7 of 12
Tensors and Exterior Algebra4 of 11
Variational Principles and Interlacing2 of 11
Computation and Applications1 of 13
What you can skip
144 of the book's 237 sections are not on this path, and skipping them costs you no proof you need.
Not needed at all: Norms and Matrix Analysis, Convexity, Nonnegative Matrices, 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: 13.13 Projections Revisited, 14.12 A First Look at Clifford Algebras, 17.4 Deleting a Row and a Column, 19.9 M-Matrices.
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.