STUDENT MATHEMATICAL LIBRARY Volume 29

Matrix Groups for Undergraduate s

Kristopher Tapp

• A M S AMERICAN MATHEMATICA L SOCIET Y

http://dx.doi.org/10.1090/stml/029

Editorial Boar d

Davide P . Cervon e Bra d Osgoo d Robin Forma n Car l Pomeranc e (Chair )

2000 Mathematics Subject Classification. P r i m a r y 20-02 , 20G20 ; Secondary 22C05 , 22E15 .

T h e ar twor k o n th e cove r i s a computer -manipu la te d pho tograp h create d by Char i t y Hendrickso n an d Kris tophe r Tapp .

For addi t iona l informatio n an d upda t e s o n th i s book , visi t w w w . a m s . o r g / b o o k p a g e s / s t m l - 2 9

Library o f Congres s Cataloging-in-Publicat io n D a t a

Tapp, Kristopher , 1971 -Matrix group s fo r undergraduate s / Kristophe r Tapp .

p. cm . - (Studen t mathematica l library , ISS N 1520-912 1 ; v. 29 ) Includes bibliographica l reference s an d index . ISBN 0-8218-3785- 0 (pbk. : acid-fre e paper ) 1. Matri x groups . 2 . Linea r algebrai c groups . 3 . Compac t groups . 4 . Li e

groups. I . Title . II . Series .

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Contents

Why stud y matri x groups ? 1

Chapter 1 . Matrice s 5

§1. Rigi d motion s o f the sphere : a motivatin g exampl e 5

§2. Field s an d skew-field s 7

§3. Th e quaternion s 8

§4. Matri x operation s 1 1

§5. Matrice s a s linea r transformation s 1 5

§6. Th e genera l linea r group s 1 7

§7. Chang e o f basi s via conjugatio n 1 8

§8. Exercise s 2 0

Chapter 2 . Al l matrix group s ar e rea l matri x group s 2 3

§1. Comple x matrice s a s rea l matrice s 2 4

§2. Quaternioni c matrice s a s comple x matrice s 2 8

§3. Restrictin g t o th e genera l linea r group s 3 0

§4. Exercise s 3 2

Chapter 3 . Th e orthogona l group s 3 3

§1. Th e standar d inne r produc t o n K n 3 3

§2. Severa l characterizations o f the orthogona l group s 3 6

iii

IV Contents

§3. Th e specia l orthogona l group s 3 9

§4. Lo w dimensiona l orthogona l group s 4 0

§5. Orthogona l matrice s an d isometrie s 4 1

§6. Th e isometr y grou p o f Euclidean spac e 4 3

§7. Symmetr y group s 4 5

§8. Exercise s 4 7

Chapter 4 . Th e topolog y o f matrix group s 5 1

§1. Ope n an d close d set s an d limi t point s 5 2

§2. Continuit y 5 7

§3. Path-connecte d set s 5 9

§4. Compac t set s 6 0

§5. Definitio n an d example s o f matrix group s 6 2

§6. Exercise s 6 4

Chapter 5 . Li e algebra s 6 7

§1. Th e Li e algebr a i s a subspace 6 8

§2. Som e example s o f Lie algebra s 7 0

§3. Li e algebra vector s a s vector fields 7 3

§4. Th e Li e algebra s o f the orthogona l group s 7 5

§5. Exercise s 7 7

Chapter 6 . Matri x exponentiatio n 7 9

§1. Serie s i n K 7 9

§2. Serie s i n M n(K) 8 2

§3. Th e bes t pat h i n a matri x grou p 8 4

§4. Propertie s o f the exponentia l ma p 8 6

§5. Exercise s 9 0

Chapter 7 . Matri x group s ar e manifold s 9 3

§1. Analysi s backgroun d 9 4

§2. Proo f o f par t (1 ) o f Theorem 7. 1 9 8

§3. Proo f o f par t (2 ) o f Theorem 7. 1 10 0

Contents v

§4.

§5-

§6-

Chapt<

§i.

§2.

§3-

§4.

§5-

§6-

§7-

§8-

Chapt<

§1.

§2.

§3.

§4.

§5.

§6.

§7.

§8.

§9-

§10.

Manifolds

More abou t manifold s

Exercises

sr 8 . Th e Li e bracke t

The Li e bracke t

The adjoin t actio n

Example: th e adjoin t actio n fo r SO(3)

The adjoin t actio n fo r compac t matri x group s

Global conclusion s

The doubl e cove r Sp(l) - + SO(3)

Other doubl e cover s

Exercises

er 9 . Maxima l tor i

Several characterization s o f a toru s

The standar d maxima l toru s an d cente r o f SO(n), SU(n), U{n) an d Sp(n)

Conjugates o f a maxima l toru s

The Li e algebr a o f a maxima l toru s

The shap e o f SO(3)

The ran k o f a compac t matri x grou p

Who commute s wit h whom ?

The classificatio n o f compact matri x group s

Lie group s

Exercises

Bibliography

Index

103

106

110

113

113

117

120

121

124

126

130

131

135

136

140

145

152

154

155

157

158

159

160

163

165

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Bibliography

1. A . Baker , Matrix Groups: an Introduction to Lie Group Theory, Springer, 2002 .

2. Cornfed , Formin , Sinai , Ergotic Theory, Springer-Verlag , 1982.

3. M . Curtis , Matrix Groups, Second Edition, Springer , 1975 , 1984 .

4. Frobenius , Journa l fu r di e Rein e un d Angewandt e Mathematik , 1878 , Vol. 84 , 1-63 .

5. J . Gallian , Contemporary Abstract Algebra, 2002 , Houghto n Miffli n Co .

6. F . Goodman, Algebra: Abstract and Concrete Stressing Symmetry, Pren -tice Hall , 2003 .

7. B . Hall , Lie Groups, Lie Algebras, and Representations, Springer , 2003 .

8. F.R . Harvey , Spinors and Calibrations, Perspective s i n Mathematics , Vol. 9 , 1990 .

9. S . Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, American Mat h Society , 2001.

10. R . Howe , Very basic Lie theory, America n Mathematica l Monthly , 9 0 (1983), 600-623 ; Correction , Amer . Math . Monthl y 9 1 (1984) , 247 .

11. W . Rossmann , Lie Groups: an Introduction Through Linear Groups, Oxford Scienc e Publications , 2002 .

12. M . Spivak , A Comprehensive Introduction to Differential Geometry, Volume 1, 1979.

13. F . Warner , Foundations of Differentiate Manifolds and Lie Groups, Springer-Verlag, 1983 .

14. J . Weeks , The Poincare dodecahedral space and the mystery of the miss-ing fluctuations, Notice s o f the AM S 5 1 (2004) , numbe r 6 , 610-619 .

163

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Index

action o f a matri x grou p o n R m , 11 8 adjoint action , 11 8 affine group , Aff n(K), 4 9 alternating group , 4 6

ball, 5 2 block-diagonal matrix , 14 1 boundary point , 5 2 bounded, 6 0

Campbell-Baker-Hausdorff series , 12 5 Cauchy sequence , 55 center o f a matri x group , 13 6 central subgroup , 13 2 chain rul e

for Euclidea n space , 9 6 for manifolds , 10 8

change o f basi s matrix , 1 9 clopen - bot h ope n an d closed , 5 9 closed set , 5 3 commutative diagram , 2 5 compact, 6 0 complex numbers , C , 8 complex structure , 2 7 complex-linear rea l matrices , 2 6 conjugate o f a quaternion , 1 0 conjugate-transpose o f a matrix , A*,

36 continuous, 5 7 convergence

absolute, 8 0 of a sequence i n R m , 5 5 of a serie s i n K , 7 9 of a serie s i n M n(K), 8 2

Cramer's rule , 1 8

dense, 5 6

derivative directional, 9 4 of a function betwee n manifolds , 10 7 of a function fro m R m t o R n, 9 5 partial, 9 4

determinant of a quaternioni c matrix , 3 1 of a rea l o r comple x matrix , 1 3

diffeomorphic, 10 3 dihedral group , 4 5 dimension

of a manifold , 10 4 of a matri x group , 7 0

discrete subgroup , 13 9 distance functio n o n R m , 5 2 division algebra , 1 1 double cove r

5p(l) - + 50(3) , 12 6 Sp(l) x Sp(l) — 50(4), 13 0 definition of , 12 6 others, 13 0

eigenvalue, 14 7 eigenvector, 14 7 Euclidean space , R m , 5 1 exceptional groups , 15 9

field, 7 Frobenius, 1 0 fundamental domain , 13 8

general linea r groups , GL n(K), 1 7 graph o f a function , 11 1

Heine-Borel Theorem , 6 1 hermitian inne r product , 3 4

165

166 Index

homeomorphism, 5 8

ideal o f a Li e algebra , 12 4 identity componen t o f a matrix group ,

110 inner produc t o n K. n, 3 4 integral curve , 8 4 inverse functio n theore m

for Euclidea n space , 9 7 for manifolds , 10 9

isometry group of Euclidean space, Isom(E n),

43 of R n, 4 1 of a matri x group , 12 4

Lie algebra , 6 8 Lie algebra homomorphism , 11 5 Lie algebra isomorphism , 11 5 Lie bracket , 11 4 Lie correspondence theorem , 12 5 Lie group, 15 9 limit point , 5 5 linear function , 1 5 log, 10 1

manifold, 10 4 matrix exponentiation , e A, 8 3 matrix group , definitio n of , 6 3

neighborhood, 5 6 norm

of a quaternion , 1 0 of a vecto r i n K n, 3 4

normalizer, 16 2

octonians, 1 1 one-parameter group , 8 9 open cover , 6 1 open set , 5 3 orientation o f E 3, 4 2 orthogonal, 3 4 orthogonal group , 0(n), 3 6 orthonormal, 3 4

parametrization, 10 4 path-connected, 5 9 Poincare dodecahedra l space , 12 9 polar decompositio n theorem , 6 6 power series , 80

quaternionic-linear comple x matrices , 28

quaternions, 9

radius o f convergence , 8 1 rank, 15 6 real projectiv e space , MP n, 12 8

reflection, 16 1 regular

element o f a Li e algebra , 16 1 element o f a matri x group , 15 7

regular solid , 4 6 Riemannian geometry , 12 3 root test , 8 1

Schwarz inequality , 3 5 skew-field, 7 skew-hermitian matrices , u(n), 7 5 skew-symmetric matrices , so(n), 7 5 skew-symplectic matrices , sp(n), 7 5 smoothness

of a function betwee n subset s o f Eu-clidean spaces , 10 3

of a function betwee n two Euclidea n spaces, 9 5

of a n isomorphis m betwee n matri x groups, 10 9

special linea r group , SL n(K), 3 9 special orthogona l group s

50(3), 5 SO(n), 3 9

special unitar y group , SU(n), 3 9 sphere, S n, 6 spin group , 13 0 sub-algebra o f a Li e algebra , 12 4 sub-convergence, 6 2 subspace, 1 4 symmetry

direct an d indirect , 4 5 group o f a set , Symm(X) , 4 5

symplectic group , Sp(n), 3 6 symplectic inne r product , 3 5

tangent bundl e o f a manifold , 11 1 tangent space , 6 7 topology

of R m, 5 4 of a subse t o f E m , 5 7

torus definition of , 13 6 in a matri x group , 14 0 maximal, 14 0 of revolution , 10 6 standard maximal , 14 2

trace o f a matrix , 1 2 transpose o f a matrix , A T , 1 2 triangle inequality , 5 2

unit tangen t bundl e of a manifold, 11 1 unitary group , U(n), 3 6 upper triangula r matrices , grou p of ,

UTn(K), 6 5

vector field, 7 3 vector space , left , 1 4

Titles i n Thi s Serie s

29 Kristophe r Tapp , Matri x group s fo r undergraduates , 200 5

28 Emmanue l Lesigne , Head s o r tails : A n introductio n t o limi t theorem s i n probability, 200 5

27 Reinhar d Illner , C . Sea n Bohun , Samanth a McCol lum , an d The a van R o o d e , Mathematica l modelling : A cas e studie s approach , 200 5

26 Rober t Hardt , Editor , Si x theme s o n variation , 200 4

25 S . V . Duzhi n an d B . D . Chebotarevsky , Transformatio n group s fo r beginners, 200 4

24 Bruc e M . Landma n an d Aaro n Robertson , Ramse y theor y o n th e

integers, 200 4

23 S . K . Lando , Lecture s o n generatin g functions , 200 3

22 Andrea s Arvanitoyeorgos , A n introductio n t o Li e group s an d th e

geometry o f homogeneou s spaces , 200 3

21 W . J . Kaczo r an d M . T . Nowak , Problem s i n mathematica l analysi s

III: Integration , 200 3

20 Klau s Hulek , Elementar y algebrai c geometry , 200 3

19 A . She n an d N . K . Vereshchagin , Computabl e functions , 200 3

18 V . V . Yaschenko , Editor , Cryptography : A n introduction , 200 2

17 A . She n an d N . K . Vereshchagin , Basi c se t theory , 200 2 16 Wolfgan g Kuhnel , Differentia l geometry : curve s - surface s - manifolds ,

2002

15 Ger d Fischer , Plan e algebrai c curves , 200 1

14 V . A . Vassiliev , Introductio n t o topology , 200 1 13 Frederic k J . Almgren , Jr. , Plateau' s problem : A n invitatio n t o varifol d

geometry, 200 1 12 W . J . Kaczo r an d M . T . Nowak , Problem s i n mathematica l analysi s

II: Continuit y an d differentiation , 200 1

11 Michae l Mester ton-Gibbons , A n introductio n t o game-theoreti c modelling, 200 0

® 10 Joh n Oprea , Th e mathematic s o f soa p films: Exploration s wit h Mapl e ,

2000

9 Davi d E . Blair , Inversio n theor y an d conforma l mapping , 200 0

8 Edwar d B . Burger , Explorin g th e numbe r jungle : A journey int o diophantine analysis , 200 0

7 Jud y L . Walker , Code s an d curves , 200 0

6 Geral d Tenenbau m an d Miche l Mende s France , Th e prim e number s and thei r distribution , 200 0

5 Alexande r Mehlmann , Th e game' s afoot ! Gam e theor y i n myt h an d paradox, 200 0

4 W . J . Kaczo r an d M . T . Nowak , Problem s i n mathematica l analysi s I: Rea l numbers , sequence s an d series , 200 0

TITLES I N THI S SERIE S

3 Roge r Knobel , A n introductio n t o th e mathematica l theor y o f waves , 2000

2 Gregor y F . Lawle r an d Leste r N . Coyle , Lecture s o n contemporar y

probability, 199 9

1 Charle s Radin , Mile s o f tiles , 199 9