Linear Algebra and Group TheoryPDF电子书下载

PART Ⅰ：DETERMINANTS AND SYSTEMS OF EQUATIONS 1

Chapter 1．Determinants and Their Properties 3

1．The Concept of a Determinant 3

2．Permutations 7

3．Basic Properties of Determinants 12

4．Calculation of Determinants 17

5．Examples 18

6．The Multiplication Theorem for Determinants 24

7．Rectangular Matrices 27

Problems 31

Chapter 2．Solution of Systems of Linear Equations 42

8．Cramer＇s Rule 42

9．The General Case 43

10．Homogeneous Systems 48

11．Linear Forms 50

12．n-Dimensional Vector Space 52

13．The Scalar Product 58

14．Geometrical Interpretation of Homogeneous Systems 60

15．Inhomogeneous Systems 63

16．The Gram Determinant．Hadamard＇s Inequality 66

17．Systems of Linear Differential Equations with Constant Coefficients 70

18．Jacobians 75

19．Implicit Functions 78

Problems 83

PART Ⅱ：MATRIX THEORY 93

Chapter 3．Linear Transformations 95

20．Coordinate Transformations in Three Dimensions 95

21．General Linear Transformations in Three Dimensions 99

22．Covariant and Contravariant Affine Vectors 106

23．The Tensor Concept 109

24．Cartesian Tensors 113

25．The n-Dimensional Case 116

26．Elements of Matrix Algebra 120

27．Eigenvalues of a Matrix．Reduction of a Matrix to Canonical Form 125

28．Unitary and Orthogonal Transformations 130

29．Schwarz＇s Inequality 135

30．Properties of the Scalar Product and Norm 137

31．The Orthogonalization Process for Vectors 138

Problems 140

Chapter 4．Quadratic Forms 149

32．Reduction of a Quadratic Form to a Sum of Squares 149

33．Multiple Roots of the Characteristic Equation 153

34．Examples 157

35．Classification of Quadratic Forms 160

36．Jacobi＇s Formula 165

37．Simultaneous Reduction of Two Quadratic Forms to Sums of Squares 166

38．Small Oscillations 168

39．Extremal Properties of the Eigenvalues of a Quadratic Form 170

40．Hermitian Matrices and Hermitian Forms 173

41．Commuting Hermitian Matrices 178

42．Reduction of Unitary Matrices to Diagonal Form 180

43．Projection Matrices 185

44．Functions of Matrices 190

Problems 193

Chapter 5．Infinite-Dimensional Spaces 201

45．Infinite-Dimensional Spaces 201

46．Convergence of Vectors 206

47．Complete Systems of Orthonormal Vectors 210

48．Linear Transformations in Infinitely Many Variables 214

49．Function Space 218

50．Relation between the Spaces F and H 221

51．Linear Operators 224

Problems 230

Chapter 6．Reduction of Matrices to Canonical Form 234

52．Preliminary Considerations 234

53．The Case of Distinct Roots 240

54．The Case of Multiple Roots．First Step in the Reduction 242

55．Reduction to Canonical Form 245

56．Determination of the Structure of the Canonical Form 251

57．An Example 254

Problems 260

PART Ⅲ：GROUP THEORY 265

Chapter 7．Elements of the General Theory of Groups 267

58．Groups of Linear Transformations 267

59．The Polyhedral Groups 270

60．Lorentz Transformations 273

61．Permutations 279

62．Abstract Groups 283

63．Subgroups 286

64．Classes and Normal Subgroups 289

65．Examples 292

66．Isomorphic and Homomorphic Groups 294

67．Examples 296

68．Stereographic Projection 298

69．The Unitary Group and the Rotation Group 300

70．The Unimodular Group and the Lorentz Group 305

Problems 309

Chapter 8．Representations of Groups 315

71．Representation of Groups by Linear Transformations 315

72．Basic Theorems 319

73．Abelian Groups and One-Dimensional Representations 323

74．Representations of the Two-Dimensional Unitary Group 325

75．Representations of the Rotation Group 331

76．Proof That the Rotation Group Is Simple 334

77．Laplace＇s Equation and Representations of the Rotation Group 336

78．The Direct Product of Two Matrices 341

79．The Direct Product of Two Representations of a Group 343

80．The Direct Product of Two Groups and its Representations 346

81．Reduction of the Direct Product Dj×Dj＇ of Two Representations of the Rotation Group 349

82．The Orthogonality Property 355

83．Characters 359

84．The Regular Representation of a Group 365

85．Examples of Representations of Finite Groups 367

86．Representations of the Two-Dimensional Unimodular Group 370

87．Proof That the Lorentz Group Is Simple 374

Problems 375

Chapter 9．Continuous Groups 381

88．Continuous Groups．Structure Constants 381

89．Infinitesimal Transformations 385

90．The Rotation Group 388

91．Infinitesimal Transformations and Representations of the Rotation Group 390

92．Representations of the Lorentz Group 394

93．Auxiliary Formulas 397

94．Construction of a Group from its Structure Constants 400

95．Integration on a Group．The Orthogonality Property 402

96．Examples 409

Problems 415

Appendix 419

Bibliography 429

Hints and Answers 431

Index 459