**Student Mathematical Library**

Volume: 79;
2016;
239 pp;
Softcover

MSC: Primary 20;
Secondary 22

**Print ISBN: 978-1-4704-2722-1
Product Code: STML/79**

List Price: $49.00

Individual Price: $39.20

**Electronic ISBN: 978-1-4704-2938-6
Product Code: STML/79.E**

List Price: $49.00

Individual Price: $39.20

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#### Supplemental Materials

# Matrix Groups for Undergraduates: Second Edition

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*Kristopher Tapp*

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.

From reviews of the First Edition:

This book could be used as an excellent textbook for a one semester course at university and it will prepare students for a graduate course on Lie groups, Lie algebras, etc. … The book combines an intuitive style of writing with rigorous definitions and proofs, giving examples from fields of mathematics, physics, and other sciences where matrices are successfully applied. The book will surely be interesting and helpful for students in algebra and their teachers.

—European Mathematical Society Newsletters

This is an excellent, well-written textbook which is strongly recommended to a wide audience of readers interested in mathematics and its applications. The book is suitable for a one semester undergraduate lecture course in matrix groups, and would also be useful supplementary reading for more general group theory courses.

—MathSciNet (or Mathematical Reviews)

#### Readership

Undergraduate and graduate students and research mathematicians interested in teaching and learning Lie groups, in particular, classical Lie groups.

#### Reviews & Endorsements

This book offers a very nice introduction to the theory of matrix groups and their Lie algebras. The background is kept to a minimum, only basics of calculus, linear algebra and group theory are assumed, while background on topology (of subsets of Euclidean space) is developed in the text. While the text gives complete and exact proofs, it is easy to read, appeals to intuition, and contains many pictures and helpful exercises.

-- A. Cap, Monatshefte für Mathematik

[T]he second edition is an expanded and improved version of the original. It can be strongly recommended for an undergraduate course in Lie groups, or as complementary reading for a course in group theory. Prerequisites are basic: knowledge of algebra, geometry, and analysis at an undergraduate level. Hence the book is suitable for a wide audience of readers who are meeting applications of group theory in other areas of mathematics and physics, or even further afield.

-- Alla S. Detinko, Mathematical Reviews

The author gives an inspiring presentation of the topics presented in this book.

-- Erich W. Ellers, Zentralblatt Math

#### Table of Contents

# Table of Contents

## Matrix Groups for Undergraduates: Second Edition

- Cover Cover11
- Title page iii4
- Why study matrix groups? 110
- Chapter 1. Matrices 514
- Chapter 2. All matrix groups are real matrix groups 2332
- Chapter 3. The orthogonal groups 3342
- 1. The standard inner product on \Kⁿ 3342
- 2. Several characterizations of the orthogonal groups 3645
- 3. The special orthogonal groups 3948
- 4. Low dimensional orthogonal groups 4049
- 5. Orthogonal matrices and isometries 4150
- 6. The isometry group of Euclidean space 4352
- 7. Symmetry groups 4554
- 8. Exercises 4857

- Chapter 4. The topology of matrix groups 5362
- Chapter 5. Lie algebras 6978
- Chapter 6. Matrix exponentiation 8190
- Chapter 7. Matrix groups are manifolds 95104
- Chapter 8. The Lie bracket 117126
- 1. The Lie bracket 117126
- 2. The adjoint representation 121130
- 3. Example: the adjoint representation for 𝑆𝑂(3) 124133
- 4. The adjoint representation for compact matrix groups 125134
- 5. Global conclusions 128137
- 6. The double cover 𝑆𝑝(1)\ra𝑆𝑂(3) 130139
- 7. Other double covers 133142
- 8. Exercises 135144

- Chapter 9. Maximal tori 139148
- 1. Several characterizations of a torus 140149
- 2. The standard maximal torus and center of 𝑆𝑂(𝑛), 𝑆𝑈(𝑛), 𝑈(𝑛) and 𝑆𝑝(𝑛) 144153
- 3. Conjugates of a maximal torus 149158
- 4. The Lie algebra of a maximal torus 156165
- 5. The shape of 𝑆𝑂(3) 157166
- 6. The rank of a compact matrix group 159168
- 7. Exercises 161170

- Chapter 10. Homogeneous manifolds 163172
- Chapter 11. Roots 197206
- 1. The structure of 𝑠𝑢(3) 198207
- 2. The structure of \mg=𝑠𝑢(𝑛) 201210
- 3. An invariant decomposition of \mg 204213
- 4. The definition of roots and dual roots 206215
- 5. The bracket of two root spaces 210219
- 6. The structure of 𝑠𝑜(2𝑛) 212221
- 7. The structure of 𝑠𝑜(2𝑛+1) 214223
- 8. The structure of 𝑠𝑝(𝑛) 215224
- 9. The Weyl group 216225
- 10. Towards the classification theorem 221230
- 11. Complexified Lie algebras 225234
- 12. Exercises 230239

- Bibliography 235244
- Index 237246
- Back Cover Back Cover1250