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Matrix Groups for Undergraduates

Kristopher Tapp
Publisher: 
American Mathematical Society
Publication Date: 
2005
Number of Pages: 
166
Format: 
Paperback
Series: 
Student Mathematical Library 29
Price: 
29.00
ISBN: 
0-8218-3785-0
Category: 
Monograph
[Reviewed by
David P. Roberts
, on
01/3/2006
]

The back cover of Matrix Groups for Undergraduates says it is a text for a one-semester upper-level undergraduate course which will assist in preparing students for graduate school. A course based on this text would indeed be helpful to many students.

Tapp's book weighs in at a very slender 169 pages. Its chapter list gives a good indication of the content and level:

 

Why study matrix groups?

5.

Lie algebras

1.

Matrices

6.

Matrix exponentiation

2.

All matrix groups are real matrix groups

7.

Matrix groups are manifolds

3.

The orthogonal groups

8.

The Lie bracket

4.

The topology of matrix groups

9.

Maximal tori


The preface answers its title question in a very enticing way: applications from graphic programming to quantum computing are mentioned. The chapters themselves mostly focus on basics, with only occasional brief revisiting of the intriguing applications.

Here are four of the ways that the author keeps the text at an undergraduate level:

1. There is considerable emphasis throughout on examples. Since the classical examples of matrix groups dominate the theory anyway, the focus on examples really has no downside. Throughout the book the author is careful to treat the reals R, the complexes C, and the quaternions H as uniformly as possible. Thus he is able to introduce orthogonal groups O(n), unitary groups U(n), and symplectic groups Sp(n) in a parallel way. Theorem 9.31, appropriately stated without proof, forms a satisfying conclusion to the course. It tells readers that the three sequences they have been intensively studying, together with five exceptional groups they have not seen, form the building blocks of all compact matrix groups.

2. There is further emphasis on low dimensions and visibility. The first chapter explains in intuitive terms why the group SO(3) of rotations of a globe is three-dimensional. It previews the idea of maximal torus by explaining, "Rotating the globe around the axis through the North and South Pole provides a 'circle's worth' of elements of SO(3)." There are helpful pictures throughout the book. Even in the second-to-last chapter, the low-dimensional isomorphism from Sp(1) to SU(2) and the double cover from SU(2) to SO(3) play a prominent role.

3. There is review appropriate to the intended readers. Aspects of linear algebra over R and C are reviewed in the process of presenting new material corresponding to H. Chapters 4 and 7 are almost entirely devoted to background material, on point-set topology and manifolds respectively; in each case, everything takes place in ambient Euclidean spaces Rn. Even in the last chapter, a theme is that diagonalization theorems of linear algebra are being revisited.

4. Each chapter concludes with approximately 15 exercises. Some are theoretical, like 4.1 through 4.6, each of which has the form "Prove Proposition 4.x." Many reinforce theoretical topics by considering them in examples, such as Exercise 6.5 which asks students to describe all one-parameter subgroups of GL1(C) and draw some in the x-y plane. The many exercises would support a course where students regularly present material in class.

Tapp writes towards the end of his preface, "I believe that matrix groups should become a more common staple of the undergraduate curriculum; my hope is that this text will help allow a movement in that direction." With his gently written text and strictly bounded goals, he has succeeded.


David Roberts is an associate professor of mathematics at the University of Minnesota, Morris.

* Why study matrix groups?
* Matrices
* All matrix groups are real matrix groups
* The orthogonal groups
* The topology of matrix groups
* Lie algebras
* Matrix exponentiation
* Matrix groups are manifolds
* The Lie bracket
* Maximal tori
* Bibliography
* Index

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