Linear Algebra and Group Theory

Cover page ......Page 1
Title page ......Page 3
Preface ......Page 5
Contents ......Page 7
PART I: DETERMINANTS AND SYSTEMS OF EQUATIONS ......Page 11
1. The Concept of a Determinant ......Page 13
2. Permutations ......Page 17
3. Basic Properties of Determinants ......Page 22
4. Calculation of Determinants ......Page 27
5. Examples ......Page 28
6. The Multiplication Theorem for Determinants ......Page 34
7. Rectangular Matrices ......Page 37
Problems ......Page 41
8. Cramer\'s Rule ......Page 52
9. The General Case ......Page 53
10. Homogeneous Systems ......Page 58
11. Linear Forms ......Page 60
12. n-Dimensional Vector Space ......Page 62
13. The Scalar Product ......Page 68
14. Geometrical Interpretation of Homogeneous Systems ......Page 70
15. Inhomogeneous Systems ......Page 73
16. The Gram Determinant. Hadamard\'s Inequality ......Page 76
17. Systems of Linear Differential Equations with Constant Coefficients ......Page 80
18. Jacobians ......Page 85
19. Implicit Functions ......Page 88
Problems ......Page 93
PART II: MATRIX THEORY ......Page 103
20. Coordinate Transformations in Three Dimensions ......Page 105
21. General Linear Transformations in Three Dimensions ......Page 109
22. Covariant and Contravariant Affine Vectors ......Page 116
23. The Tensor Concept ......Page 119
24. Cartesian Tensors ......Page 123
25. The n-Dimensional Case ......Page 126
26. Elements of Matrix Algebra ......Page 130
27. Eigenvalues of a Matrix. Reduction of a Matrix to Canonical Form ......Page 135
28. Unitary and Orthogonal Transformations ......Page 140
29. Schwarz\'s Inequality ......Page 145
30. Properties of the Scalar Product and Norm ......Page 147
31. The Orthogonalization Process for Vectors ......Page 148
Problems ......Page 150
32. Reduction of a Quadratic Form to a Sum of Squares ......Page 159
33. Multiple Roots of the Characteristic Equation ......Page 163
34. Examples ......Page 167
35. Classification of Quadratic Forms ......Page 170
36. Jacobi\'s Formula ......Page 175
37. Simultaneous Reduction of Two Quadratic Forms to Sums of Squares ......Page 176
38. Small Oscillations ......Page 178
39. Extremal Properties of the Eigenvalues of\'a Quadratic Form ......Page 180
40. Hermitian Matrices and Hermitian Forms ......Page 183
41. Commuting Hermitian Matrices ......Page 188
42. Reduction of Unitary Matrices to Diagonal Form ......Page 190
43. Projection Matrices ......Page 195
44. Functions of Matrices ......Page 200
Problems ......Page 203
45. Infinite-Dimensional Spaces ......Page 211
46. Convergence of Vectors ......Page 216
47. Complete Systems of Orthonormal Vectors ......Page 220
48. Linear Transformations in Infinitely Many Variables ......Page 224
49. Function Space ......Page 228
50. Relation between the Spaces F and H ......Page 231
51. Linear Operators ......Page 234
Problems ......Page 240
52. Preliminary Considerations ......Page 244
53. The Case of Distinct Roots ......Page 250
54. The Case of Multiple Roots First Step in the Reduction ......Page 252
55. Reduction to Canomical Form ......Page 255
56. Determination of the Structure of the Canonical Form ......Page 261
57. An Example ......Page 264
Problems ......Page 270
PART III: GROUP THEORY ......Page 275
58. Groups of Linear Transformations ......Page 277
59. The Polyhedral Groups ......Page 280
60. Lorentz Transformations ......Page 283
61. Permutations ......Page 289
62. Abstract Groups ......Page 293
63. Subgroups ......Page 296
64. Classes and Normal Subgroups ......Page 299
65. Examples ......Page 302
66. Isomorphic and Homomorphic Groups ......Page 304
67. Examples ......Page 306
68. Stereographic Projection ......Page 308
69. The Unitary Group and the Rotation Group ......Page 310
70. The Unimodular Group and the Lorentz Group ......Page 315
Problems ......Page 319
71. Representation of Groups by Linear Transformations ......Page 325
72. Basic Theorems ......Page 329
73. AbelJan Groups and One-Dimensional Representations ......Page 333
74. Representations of the Two-Dimensional Unitary Group ......Page 335
75. Representations of the Rotation Group ......Page 341
76. Proof That the Rotation Group Is Simple ......Page 344
77. Laplace\'s Equation and Representations of the Rotation Group ......Page 346
78. The Direct Product of Two Matrices ......Page 351
79. The Direct Product of Two Representations of a Group ......Page 353
80. The Direct Product of Two Groups and its Representations ......Page 356
81. Reduction of the Direct Product D_j × D_j\' of Two Representations of the Rotation Group ......Page 359
82. The Orthogonality Property ......Page 365
83. Characters ......Page 369
84. The Regular Representation of a Group ......Page 375
85. Examples of Representations of Finite Groups ......Page 377
86. Representations of the Two-Dimensional Unimodular Group ......Page 380
87. Proof That the Lorentz Group Is Simple ......Page 384
Problems ......Page 385
88. Continuous Groups. Structure Constants ......Page 391
89. Infinitesimal Transformations ......Page 395
90. The Rotation Group ......Page 398
91. Infiniteshna! Transformations and Representations of the Rotation Group ......Page 400
92. Representations of the Lorentz Group ......Page 404
93. Auxiliary Formulas ......Page 407
94. Construction of a Group from its Structure Constants ......Page 410
95. Integration on a Group. The Orthogonality Property ......Page 412
96. Examples ......Page 419
Problems ......Page 425
Appendix ......Page 429
Bibliography ......Page 439
Hints and Answers ......Page 441
Index ......Page 469