Matrix groups for undergraduates:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, Rhode Island
American Mathematical Society
2016
|
| Edition: | Second edition |
| Series: | Student mathematical library
volume 79 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | viii, 239 Seiten Illustrationen |
| ISBN: | 9781470427221 |
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Record in the Search Index
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| adam_text | Titel: Matrix groups for undergraduates
Autor: Tapp, Kristopher
Jahr: 2016
Contents
Why study matrix groups? 1
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
v
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 M„(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),
SJJ{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 ¿ 0(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
|
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| dewey-full | 512/.2 |
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| dewey-ones | 512 - Algebra |
| dewey-raw | 512/.2 |
| dewey-search | 512/.2 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | Second edition |
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| illustrated | Illustrated |
| indexdate | 2024-07-10T07:29:11Z |
| institution | BVB |
| isbn | 9781470427221 |
| language | English |
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| physical | viii, 239 Seiten Illustrationen |
| publishDate | 2016 |
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| publisher | American Mathematical Society |
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| series | Student mathematical library |
| series2 | Student mathematical library |
| spelling | Tapp, Kristopher 1971- (DE-588)124296092 aut Matrix groups for undergraduates Kristopher Tapp Second edition Providence, Rhode Island American Mathematical Society 2016 viii, 239 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Student mathematical library volume 79 Groupes compacts Groupes linéaires algébriques Lie, Groupes de Matrices, Groupes de Matrix groups Linear algebraic groups Compact groups Lie groups Matrizengruppe (DE-588)4169127-1 gnd rswk-swf Lie-Gruppe (DE-588)4035695-4 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Matrizengruppe (DE-588)4169127-1 s Lie-Gruppe (DE-588)4035695-4 s DE-604 Student mathematical library volume 79 (DE-604)BV013184751 79 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028987713&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Tapp, Kristopher 1971- Matrix groups for undergraduates Student mathematical library Groupes compacts Groupes linéaires algébriques Lie, Groupes de Matrices, Groupes de Matrix groups Linear algebraic groups Compact groups Lie groups Matrizengruppe (DE-588)4169127-1 gnd Lie-Gruppe (DE-588)4035695-4 gnd |
| subject_GND | (DE-588)4169127-1 (DE-588)4035695-4 (DE-588)4123623-3 |
| title | Matrix groups for undergraduates |
| title_auth | Matrix groups for undergraduates |
| title_exact_search | Matrix groups for undergraduates |
| title_full | Matrix groups for undergraduates Kristopher Tapp |
| title_fullStr | Matrix groups for undergraduates Kristopher Tapp |
| title_full_unstemmed | Matrix groups for undergraduates Kristopher Tapp |
| title_short | Matrix groups for undergraduates |
| title_sort | matrix groups for undergraduates |
| topic | Groupes compacts Groupes linéaires algébriques Lie, Groupes de Matrices, Groupes de Matrix groups Linear algebraic groups Compact groups Lie groups Matrizengruppe (DE-588)4169127-1 gnd Lie-Gruppe (DE-588)4035695-4 gnd |
| topic_facet | Groupes compacts Groupes linéaires algébriques Lie, Groupes de Matrices, Groupes de Matrix groups Linear algebraic groups Compact groups Lie groups Matrizengruppe Lie-Gruppe Lehrbuch |
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