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." "M...
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Providence, RI
American Math. Soc.
2005
|
| Schriftenreihe: | Student mathematical library
29 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Zusammenfassung: | "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 tigor and intuition to describe basic objects of Lie theory, Lie algebras, matrix exponentiation, Lie brackets, and maximal tori."--BOOK JACKET. |
| Beschreibung: | Includes bibliographical references (p. 163) and index |
| Beschreibung: | V, 166 S. Ill., graph. Darst. |
| ISBN: | 0821837850 9780821837856 |
Internformat
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| 490 | 1 | |a Student mathematical library |v 29 | |
| 500 | |a Includes bibliographical references (p. 163) and index | ||
| 520 | 1 | |a "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 tigor and intuition to describe basic objects of Lie theory, Lie algebras, matrix exponentiation, Lie brackets, and maximal tori."--BOOK JACKET. | |
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| 650 | 4 | |a Groupes linéaires algébriques | |
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| 650 | 4 | |a Matrices, Groupes de | |
| 650 | 4 | |a Matrix groups | |
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Datensatz im Suchindex
| _version_ | 1804133578867474432 |
|---|---|
| adam_text | 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 32
Chapter 3. The orthogonal groups 33
§1. The standard inner product on K™ 33
§2. Several characterizations of the orthogonal groups 36
iii
iv 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 47
Chapter 4. The topology of matrix groups 51
§1. Open and closed sets and limit points 52
§2. Continuity 57
§3. Path connected sets 59
§4. Compact sets 60
§5. Definition and examples of matrix groups 62
§6. Exercises 64
Chapter 5. Lie algebras 67
§1. The Lie algebra is a subspace 68
§2. Some examples of Lie algebras 70
§3. Lie algebra vectors as vector fields 73
§4. The Lie algebras of the orthogonal groups 75
§5. Exercises 77
Chapter 6. Matrix exponentiation 79
§1. Series in K 79
§2. Series in Mn(K) 82
§3. The best path in a matrix group 84
§4. Properties of the exponential map 86
§5. Exercises 90
Chapter 7. Matrix groups are manifolds 93
§1. Analysis background 94
§2. Proof of part (1) of Theorem 7.1 98
§3. Proof of part (2) of Theorem 7.1 100
Contents v
§4. Manifolds 103
§5. More about manifolds 106
§6. Exercises 110
Chapter 8. The Lie bracket 113
§1. The Lie bracket 113
§2. The adjoint action 117
§3. Example: the adjoint action for SO{3) 120
§4. The adjoint action for compact matrix groups 121
§5. Global conclusions 124
§6. The double cover Sp(l) SO(3) 126
§7. Other double covers 130
§8. Exercises 131
Chapter 9. Maximal tori 135
§1. Several characterizations of a torus 136
§2. The standard maximal torus and center of SO(n),
SU{n), U{n) and Sp(n) 140
§3. Conjugates of a maximal torus 145
§4. The Lie algebra of a maximal torus 152
§5. The shape of SO(3) 154
§6. The rank of a compact matrix group 155
§7. Who commutes with whom? 157
§8. The classification of compact matrix groups 158
§9. Lie groups 159
§10. Exercises 160
Bibliography 163
Index 165
|
| any_adam_object | 1 |
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| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV020018456 |
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| indexdate | 2024-07-09T20:10:57Z |
| institution | BVB |
| isbn | 0821837850 9780821837856 |
| language | English |
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| spelling | Tapp, Kristopher 1971- Verfasser (DE-588)124296092 aut Matrix groups for undergraduates Kristopher Tapp Providence, RI American Math. Soc. 2005 V, 166 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Student mathematical library 29 Includes bibliographical references (p. 163) and index "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 tigor and intuition to describe basic objects of Lie theory, Lie algebras, matrix exponentiation, Lie brackets, and maximal tori."--BOOK JACKET. 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 29 (DE-604)BV013184751 29 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013339884&sequence=000002&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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