Matrix Groups for Undergraduates

Matrix groups touch an enormous spectrum of the mathematical arena. This textbook brings them into the undergraduate curriculum. It makes an excellent one-semester course for students familiar with linear and abstract algebra and prepares them for a graduate course on Lie groups.

Matrix Groups for Undergraduates is concrete and example-driven, with geometric motivation and rigo...

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Matrix groups touch an enormous spectrum of the mathematical arena. This textbook brings them into the undergraduate curriculum. It makes an excellent one-semester course for students familiar with linear and abstract algebra and prepares them for a graduate course on Lie groups.

Matrix Groups for Undergraduates is concrete and example-driven, with geometric motivation and rigorous proofs. The story begins and ends with the rotations of a globe. In between, the author combines rigor and intuition to describe the basic objects of Lie theory: Lie algebras, matrix exponentiation, Lie brackets, maximal tori, homogeneous spaces, and roots.

This second edition includes two new chapters that allow for an easier transition to the general theory of Lie groups.

Cover 1
Title page 4
Why study matrix groups? 10
Chapter 1. Matrices 14
1. Rigid motions of the sphere: a motivating example 14
2. Fields and skew-fields 16
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Cover 1
Title page 4
Why study matrix groups? 10
Chapter 1. Matrices 14
1. Rigid motions of the sphere: a motivating example 14
2. Fields and skew-fields 16
3. The quaternions 17
4. Matrix operations 20
5. Matrices as linear transformations 24
6. The general linear groups 26
7. Change of basis via conjugation 27
8. Exercises 29
Chapter 2. All matrix groups are real matrix groups 32
1. Complex matrices as real matrices 33
2. Quaternionic matrices as complex matrices 37
3. Restricting to the general linear groups 39
4. Exercises 40
Chapter 3. The orthogonal groups 42
1. The standard inner product on \Kⁿ 42
2. Several characterizations of the orthogonal groups 45
3. The special orthogonal groups 48
4. Low dimensional orthogonal groups 49
5. Orthogonal matrices and isometries 50
6. The isometry group of Euclidean space 52
7. Symmetry groups 54
8. Exercises 57
Chapter 4. The topology of matrix groups 62
1. Open and closed sets and limit points 63
2. Continuity 68
3. Path-connected sets 70
4. Compact sets 71
5. Definition and examples of matrix groups 73
6. Exercises 75
Chapter 5. Lie algebras 78
1. The Lie algebra is a subspace 79
2. Some examples of Lie algebras 81
3. Lie algebra vectors as vector fields 84
4. The Lie algebras of the orthogonal groups 86
5. Exercises 88
Chapter 6. Matrix exponentiation 90
1. Series in \K 90
2. Series in 𝑀_{𝑛}(\K) 93
3. The best path in a matrix group 95
4. Properties of the exponential map 97
5. Exercises 101
Chapter 7. Matrix groups are manifolds 104
1. Analysis background 105
2. Proof of part (1) of Theorem 7.1 109
3. Proof of part (2) of Theorem 7.1 111
4. Manifolds 114
5. More about manifolds 117
6. Exercises 121
Chapter 8. The Lie bracket 126
1. The Lie bracket 126
2. The adjoint representation 130
3. Example: the adjoint representation for 𝑆𝑂(3) 133
4. The adjoint representation for compact matrix groups 134
5. Global conclusions 137
6. The double cover 𝑆𝑝(1)\ra𝑆𝑂(3) 139
7. Other double covers 142
8. Exercises 144
Chapter 9. Maximal tori 148
1. Several characterizations of a torus 149
2. The standard maximal torus and center of 𝑆𝑂(𝑛), 𝑆𝑈(𝑛), 𝑈(𝑛) and 𝑆𝑝(𝑛) 153
3. Conjugates of a maximal torus 158
4. The Lie algebra of a maximal torus 165
5. The shape of 𝑆𝑂(3) 166
6. The rank of a compact matrix group 168
7. Exercises 170
Chapter 10. Homogeneous manifolds 172
1. Generalized manifolds 172
2. The projective spaces 178
3. Coset spaces are manifolds 181
4. Group actions 184
5. Homogeneous manifolds 186
6. Riemannian manifolds 191
7. Lie groups 196
8. Exercises 201
Chapter 11. Roots 206
1. The structure of 𝑠𝑢(3) 207
2. The structure of \mg=𝑠𝑢(𝑛) 210
3. An invariant decomposition of \mg 213
4. The definition of roots and dual roots 215
5. The bracket of two root spaces 219
6. The structure of 𝑠𝑜(2𝑛) 221
7. The structure of 𝑠𝑜(2𝑛+1) 223
8. The structure of 𝑠𝑝(𝑛) 224
9. The Weyl group 225
10. Towards the classification theorem 230
11. Complexified Lie algebras 234
12. Exercises 239
Bibliography 244
Index 246
Back Cover 250
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