Cover 1
Title 2
Copyright 3
Contents 4
Why study matrix groups? 8
Chapter 1. Matrices 12
§1. Rigid motions of the sphere: a motivating example 12
§2. Fields and skew-fields 14
§3. The quaternions 15
§4. Matrix operations 18
§5. Matrices as linear transformations 22
§6. The general linear groups 24
§7. Change of basis via conjugation 25
§8. Exercises 27
Chapter 2. All matrix groups are real matrix groups 30
§1. Complex matrices as real matrices 31
§2. Quaternionic matrices as complex matrices 35
§3. Restricting to the general linear groups 37
§4. Exercises 39
Chapter 3. The orthogonal groups 40
§1. The standard inner product on K[sup(n)] 40
§2. Several characterizations of the orthogonal groups 43
§3. The special orthogonal groups 46
§4. Low dimensional orthogonal groups 47
§5. Orthogonal matrices and isometries 48
§6. The isometry group of Euclidean space 50
§7. Symmetry groups 52
§8. Exercises 54
Chapter 4. The topology of matrix groups 58
§1. Open and closed sets and limit points 59
§2. Continuity 64
§3. Path-connected sets 66
§4. Compact sets 67
§5. Definition and examples of matrix groups 69
§6. Exercises 71
Chapter 5. Lie algebras 74
§1. The Lie algebra is a subspace 75
§2. Some examples of Lie algebras 77
§3. Lie algebra vectors as vector fields 80
§4. The Lie algebras of the orthogonal groups 82
§5. Exercises 84
Chapter 6. Matrix exponentiation 86
§1. Series in K 86
§2. Series in M[sub(n)](K) 89
§3. The best path in a matrix group 91
§4. Properties of the exponential map 93
§5. Exercises 97
Chapter 7. Matrix groups are manifolds 100
§1. Analysis background 101
§2. Proof of part (1) of Theorem 7.1 105
§3. Proof of part (2) of Theorem 7.1 107
§4. Manifolds 110
§5 More about manifolds 113
§6. Exercises 117
Chapter 8. The Lie bracket 120
§1. The Lie bracket 120
§2. The adjoint action 124
§3. Example: the adjoint action for SO(3) 127
§4. The adjoint action for compact matrix groups 128
§5. Global conclusions 131
§6. The double cover Sp(1) → SO(3) 133
§7. Other double covers 137
§8. Exercises 138
Chapter 9. Maximal tori 142
§1. Several characterizations of a torus 143
§2. The standard maximal torus and center of SO(n), SU(n), U(n) and Sp(n) 147
§3. Conjugates of a maximal torus 152
§4. The Lie algebra of a maximal torus 159
§5. The shape of SO(3) 161
§6. The rank of a compact matrix group 162
§7. Who commutes with whom? 164
§8. The classification of compact matrix groups 165
§9. Lie groups 166
§10. Exercises 167
Bibliography 170
Index 172
A 172
B 172
C 172
D 172
E 172
F 172
G 172
H 172
I 173
L 173
M 173
N 173
O 173
P 173
Q 173
R 173
S 173
T 173
U 173
V 173
Back Cover 176
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