Matrix Groups For Undergraduates

by Tapp, Kristopher

• ISBN13: 9780821837856
• ISBN10: 0821837850
• Format: Paperback
• Copyright: 2005-06-01
• Publisher: Amer Mathematical Society

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