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Linear Algebra and Group Theorypdf电子书版本下载
- 著
- 出版社： Inc.
- ISBN：
- 出版时间：1961
- 标注页数：464页
- 文件大小：90MB
- 文件页数：474页
- 主题词：
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图书目录
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