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.

**Kristopher Tapp**, Department of Math, Saint Joseph's University, Philadelphia

Chapter 1. Matrices 5

1. Rigid motions of the sphere: a motivating example 5

2. Fields and skew-fields 7

3. The quaternions 8

4. Matrix operations 11

5. Matrices as linear transformations 15

6. The general linear groups 17

7. Change of basis via conjugation 18

8. Exercises 20

Chapter 2. All matrix groups are real matrix groups 23

1. Complex matrices as real matrices 24

2. Quaternionic matrices as complex matrices 28

3. Restricting to the general linear groups 30

4. Exercises 31

Chapter 3. The orthogonal groups 33

1. The standard inner product on Kn 33

2. Several characterizations of the orthogonal groups 36

vi Contents

3. The special orthogonal groups 39

4. Low dimensional orthogonal groups 40

5. Orthogonal matrices and isometries 41

6. The isometry group of Euclidean space 43

7. Symmetry groups 45

8. Exercises 48

Chapter 4. The topology of matrix groups 53

1. Open and closed sets and limit points 54

2. Continuity 59

3. Path-connected sets 61

4. Compact sets 62

5. Definition and examples of matrix groups 64

6. Exercises 66

Chapter 5. Lie algebras 69

1. The Lie algebra is a subspace 70

2. Some examples of Lie algebras 72

3. Lie algebra vectors as vector fields 75

4. The Lie algebras of the orthogonal groups 77

5. Exercises 79

Chapter 6. Matrix exponentiation 81

1. Series in K 81

2. Series in Mn(K) 84

3. The best path in a matrix group 86

4. Properties of the exponential map 88

5. Exercises 92

Chapter 7. Matrix groups are manifolds 95

1. Analysis background 96

2. Proof of part (1) of Theorem 7.1 100

3. Proof of part (2) of Theorem 7.1 102

Contents vii

4. Manifolds 105

5. More about manifolds 108

6. Exercises 112

Chapter 8. The Lie bracket 117

1. The Lie bracket 117

2. The adjoint representation 121

3. Example: the adjoint representation for SO(3) 124

4. The adjoint representation for compact matrix groups 125

5. Global conclusions 128

6. The double cover Sp(1) ? SO(3) 130

7. Other double covers 133

8. Exercises 135

Chapter 9. Maximal tori 139

1. Several characterizations of a torus 140

2. The standard maximal torus and center of SO(n),

SU(n), U(n) and Sp(n) 144

3. Conjugates of a maximal torus 149

4. The Lie algebra of a maximal torus 156

5. The shape of SO(3) 157

6. The rank of a compact matrix group 159

7. Exercises 161

Chapter 10. Homogeneous manifolds 163

1. Generalized manifolds 163

2. The projective spaces 169

3. Coset spaces are manifolds 172

4. Group actions 175

5. Homogeneous manifolds 177

6. Riemannian manifolds 182

7. Lie groups 187

8. Exercises 192

viii Contents

Chapter 11. Roots 197

1. The structure of su(3) 198

2. The structure of g = su(n) 201

3. An invariant decomposition of g 204

4. The definition of roots and dual roots 206

5. The bracket of two root spaces 210

6. The structure of so(2n) 212

7. The structure of so(2n + 1) 214

8. The structure of sp(n) 215

9. The Weyl group 216

10. Towards the classification theorem 221

11. Complexified Lie algebras 225

12. Exercises 230

Bibliography 235

Index 237