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Matrix Theory
1
Vector spaces
Overview
1.1
Fields
1.2
Vector spaces
1.3
Lists, linear combinations, and span
1.4
Subspaces
1.5
Linear independence
1.6
Bases and coordinates
1.7
Proof patterns
2
Linear maps and matrices
Overview
2.1
Linear maps
2.2
Linear maps are determined by a basis
2.3
Matrices of linear maps
2.4
Matrix multiplication
2.5
Invertibility
2.6
Kernel and image
2.7
The Fundamental Theorems
2.8
Proof patterns
3
Linear systems and row reduction
Overview
3.1
Linear systems
3.2
Elementary matrices
3.3
Gauss–Jordan elimination
3.4
Solving linear systems
3.5
Rank and the Fundamental Theorems
3.6
Proof patterns
4
Dimension
Overview
4.1
Dimension
4.2
Rank–nullity
4.3
Rank inequalities and the transpose
4.4
Sums and direct sums
4.5
Isomorphisms and change of basis
4.6
Products and quotients
*
4.7
Affine geometry
*
4.8
Proof patterns
5
Determinants
Overview
5.1
What a determinant should be
5.2
Permutations
5.3
Existence and uniqueness
5.4
Determinants and invertibility
5.5
Cofactor expansion
5.6
Proof patterns
6
Inner product spaces
Overview
6.1
Inner products and norms
6.2
Cauchy–Schwarz and the triangle inequality
6.3
Orthogonality and orthonormal bases
6.4
Orthogonal projection and Gram–Schmidt
6.5
Least squares
6.6
Adjoints
6.7
Proof patterns
7
Eigenvalues and diagonalization
Overview
7.1
Polynomials
7.2
Eigenvalues, eigenvectors, and invariant subspaces
7.3
The characteristic polynomial
7.4
Diagonalization
7.5
Upper-triangular form and Schur’s theorem
7.6
Minimal polynomials, Cayley–Hamilton, and commuting operators
*
7.7
Proof patterns
8
The spectral theorems
Overview
8.1
Self-adjoint and normal operators
8.2
The spectral theorems
8.3
Positive operators
8.4
Isometries and unitary operators
8.5
Proof patterns
9
The singular value decomposition
Overview
9.1
Singular values and the SVD
9.2
Consequences of the SVD
*
9.3
Matrix factorizations
9.4
Proof patterns
10
Generalized eigenspaces and Jordan form
Overview
10.1
Generalized eigenvectors and nilpotent operators
10.2
The generalized eigenspace decomposition
10.3
Jordan canonical form
10.4
Proof patterns
11
Markov chains and positive matrices
Overview
11.1
Stochastic matrices and Markov chains
11.2
Perron–Frobenius
11.3
Reversibility
11.4
Proof patterns
12
Bilinear and quadratic forms
Overview
12.1
Bilinear forms
*
12.2
Quadratic forms
12.3
Positive definiteness and min–max
12.4
Inner products beyond ℝ and ℂ
*
12.5
Proof patterns
13
Duality and multilinear algebra
*
Overview
13.1
Dual spaces
13.2
Duality in inner product spaces
13.3
Multilinear maps and tensors
13.4
Alternating forms and the determinant revisited
13.5
Proof patterns
14
Linear algebra and abstract algebra
*
Overview
14.1
Groups through matrices
14.2
Fields and field extensions
14.3
Finite fields
14.4
Modules over F[x] and canonical forms
14.5
Proof patterns
15
Infinite-dimensional vector spaces
*
Overview
15.1
Bases in infinite dimensions
15.2
Linear algebra inside infinite-dimensional spaces
15.3
Inner product spaces in infinite dimensions
15.4
Proof patterns
16
Applications to data and machine learning
*
Overview
16.1
Regression, revisited
16.2
Principal component analysis
16.3
Low-rank models
16.4
Optimization of quadratic objectives
16.5
Graphs and spectral methods
16.6
Linear layers
A
Sets, functions, and proofs
Overview
A.1
Sets and functions
A.2
Induction and well-ordering
A.3
Equivalence relations
A.4
Zorn’s lemma
*
B
Complex numbers
Overview
B.1
Complex numbers
C
The Fundamental Theorem of Algebra
*
Overview
C.1
Proofs of the Fundamental Theorem of Algebra
Index of techniques
Practice
PDF
Chapter 13: Duality and multilinear algebra
13.1
Dual spaces
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13
Duality and multilinear algebra
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13.2
Duality in inner product spaces