9780821837856

Matrix Groups For Undergraduates

by
  • ISBN13:

    9780821837856

  • ISBN10:

    0821837850

  • Format: Paperback
  • Copyright: 2005-06-01
  • Publisher: Amer Mathematical Society

Note: Supplemental materials are not guaranteed with Rental or Used book purchases.

Purchase Benefits

  • Free Shipping On Orders Over $35!
    Your order must be $35 or more to qualify for free economy shipping. Bulk sales, PO's, Marketplace items, eBooks and apparel do not qualify for this offer.
  • Get Rewarded for Ordering Your Textbooks! Enroll Now
List Price: $34.00 Save up to $3.40
  • Rent Book $30.60
    Add to Cart Free Shipping

    TERM
    PRICE
    DUE

Supplemental Materials

What is included with this book?

  • The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any access cards, study guides, lab manuals, CDs, etc.
  • The Rental copy of this book is not guaranteed to include any supplemental materials. Typically, only the book itself is included. This is true even if the title states it includes any access cards, study guides, lab manuals, CDs, etc.

Summary

Matrix groups are a beautiful subject and are central to many fields in mathematics and physics. They touch upon an enormous spectrum within the mathematical arena. This textbook brings them into the undergraduate curriculum. It is excellent for a 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 basic objects of Lie theory: Lie algebras, matrix exponentiation, Lie brackets, and maximal tori. The volume is suitable for graduate students and researchers interested in group theory.

Table of Contents

Why study matrix groups? 1(4)
Matrices
5(18)
Rigid motions of the sphere: a motivating example
5(2)
Fields and skew-fields
7(1)
The quaternions
8(3)
Matrix operations
11(4)
Matrices as linear transformations
15(2)
The general linear groups
17(1)
Change of basis via conjugation
18(2)
Exercises
20(3)
All matrix groups are real matrix groups
23(10)
Complex matrices as real matrices
24(4)
Quaternionic matrices as complex matrices
28(2)
Restricting to the general linear groups
30(2)
Exercises
32(1)
The orthogonal groups
33(18)
The standard inner product on Kn
33(3)
Several characterizations of the orthogonal groups
36(3)
The special orthogonal groups
39(1)
Low dimensional orthogonal groups
40(1)
Orthogonal matrices and isometries
41(2)
The isometry group of Euclidean space
43(2)
Symmetry groups
45(2)
Exercises
47(4)
The topology of matrix groups
51(16)
Open and closed sets and limit points
52(5)
Continuity
57(2)
Path-connected sets
59(1)
Compact sets
60(2)
Definition and examples of matrix groups
62(2)
Exercises
64(3)
Lie algebras
67(12)
The Lie algebra is a subspace
68(2)
Some examples of Lie algebras
70(3)
Lie algebra vectors as vector fields
73(2)
The Lie algebras of the orthogonal groups
75(2)
Exercises
77(2)
Matrix exponentiation
79(14)
Series in K
79(3)
Series in Mn (K)
82(2)
The best path in a matrix group
84(2)
Properties of the exponential map
86(4)
Exercises
90(3)
Matrix groups are manifolds
93(20)
Analysis background
94(4)
Proof of part (1) of Theorem 7.1
98(2)
Proof of part (2) of Theorem 7.1
100(3)
Manifolds
103(3)
More about manifolds
106(4)
Exercises
110(3)
The Lie bracket
113(22)
The Lie bracket
113(4)
The adjoint action
117(3)
Example: the adjoint action for SO(3)
120(1)
The adjoint action for compact matrix groups
121(3)
Global conclusions
124(2)
The double cover Sp(1) → SO(3)
126(4)
Other double covers
130(1)
Exercises
131(4)
Maximal tori
135(28)
Several characterizations of a torus
136(4)
The standard maximal torus and center of SO(n), SU(n), U(n) and Sp(n)
140(5)
Conjugates of a maximal torus
145(7)
The Lie algebra of a maximal torus
152(2)
The shape of SO(3)
154(1)
The rank of a compact matrix group
155(2)
Who commutes with whom?
157(1)
The classification of compact matrix groups
158(1)
Lie groups
159(1)
Exercises
160(3)
Bibliography 163(2)
Index 165

Rewards Program

Write a Review