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PART Ⅰ：DETERMINANTS AND SYSTEMS OF EQUATIONS1

Chapter 1．Determinants and Their Properties3

1．The Concept of a Determinant3

2．Permutations7

3．Basic Properties of Determinants12

4．Calculation of Determinants17

5．Examples18

6．The Multiplication Theorem for Determinants24

7．Rectangular Matrices27

Problems31

Chapter 2．Solution of Systems of Linear Equations42

8．Cramer＇s Rule42

9．The General Case43

10．Homogeneous Systems48

11．Linear Forms50

12．n-Dimensional Vector Space52

13．The Scalar Product58

14．Geometrical Interpretation of Homogeneous Systems60

15．Inhomogeneous Systems63

16．The Gram Determinant．Hadamard＇s Inequality66

17．Systems of Linear Differential Equations with Constant Coefficients70

18．Jacobians75

19．Implicit Functions78

Problems83

PART Ⅱ：MATRIX THEORY93

Chapter 3．Linear Transformations95

20．Coordinate Transformations in Three Dimensions95

21．General Linear Transformations in Three Dimensions99

22．Covariant and Contravariant Affine Vectors106

23．The Tensor Concept109

24．Cartesian Tensors113

25．The n-Dimensional Case116

26．Elements of Matrix Algebra120

27．Eigenvalues of a Matrix．Reduction of a Matrix to Canonical Form125

28．Unitary and Orthogonal Transformations130

29．Schwarz＇s Inequality135

30．Properties of the Scalar Product and Norm137

31．The Orthogonalization Process for Vectors138

Problems140

Chapter 4．Quadratic Forms149

32．Reduction of a Quadratic Form to a Sum of Squares149

33．Multiple Roots of the Characteristic Equation153

34．Examples157

35．Classification of Quadratic Forms160

36．Jacobi＇s Formula165

37．Simultaneous Reduction of Two Quadratic Forms to Sums of Squares166

38．Small Oscillations168

39．Extremal Properties of the Eigenvalues of a Quadratic Form170

40．Hermitian Matrices and Hermitian Forms173

41．Commuting Hermitian Matrices178

42．Reduction of Unitary Matrices to Diagonal Form180

43．Projection Matrices185

44．Functions of Matrices190

Problems193

Chapter 5．Infinite-Dimensional Spaces201

45．Infinite-Dimensional Spaces201

46．Convergence of Vectors206

47．Complete Systems of Orthonormal Vectors210

48．Linear Transformations in Infinitely Many Variables214

49．Function Space218

50．Relation between the Spaces F and H221

51．Linear Operators224

Problems230

Chapter 6．Reduction of Matrices to Canonical Form234

52．Preliminary Considerations234

53．The Case of Distinct Roots240

54．The Case of Multiple Roots．First Step in the Reduction242

55．Reduction to Canonical Form245

56．Determination of the Structure of the Canonical Form251

57．An Example254

Problems260

PART Ⅲ：GROUP THEORY265

Chapter 7．Elements of the General Theory of Groups267

58．Groups of Linear Transformations267

59．The Polyhedral Groups270

60．Lorentz Transformations273

61．Permutations279

62．Abstract Groups283

63．Subgroups286

64．Classes and Normal Subgroups289

65．Examples292

66．Isomorphic and Homomorphic Groups294

67．Examples296

68．Stereographic Projection298

69．The Unitary Group and the Rotation Group300

70．The Unimodular Group and the Lorentz Group305

Problems309

Chapter 8．Representations of Groups315

71．Representation of Groups by Linear Transformations315

72．Basic Theorems319

73．Abelian Groups and One-Dimensional Representations323

74．Representations of the Two-Dimensional Unitary Group325

75．Representations of the Rotation Group331

76．Proof That the Rotation Group Is Simple334

77．Laplace＇s Equation and Representations of the Rotation Group336

78．The Direct Product of Two Matrices341

79．The Direct Product of Two Representations of a Group343

80．The Direct Product of Two Groups and its Representations346

81．Reduction of the Direct Product Dj×Dj＇ of Two Representations of the Rotation Group349

82．The Orthogonality Property355

83．Characters359

84．The Regular Representation of a Group365

85．Examples of Representations of Finite Groups367

86．Representations of the Two-Dimensional Unimodular Group370

87．Proof That the Lorentz Group Is Simple374

Problems375

Chapter 9．Continuous Groups381

88．Continuous Groups．Structure Constants381

89．Infinitesimal Transformations385

90．The Rotation Group388

91．Infinitesimal Transformations and Representations of the Rotation Group390

92．Representations of the Lorentz Group394

93．Auxiliary Formulas397

94．Construction of a Group from its Structure Constants400

95．Integration on a Group．The Orthogonality Property402

96．Examples409

Problems415

Appendix419

Bibliography429

Hints and Answers431

Index459