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线性代数第二课-A second course in linear algebra

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标题(title):A second course in linear algebra
线性代数第二课
作者(author):Garcia, Stephan Ramon; Horn, Roger A.
出版社(publisher):Cambridge University Press
大小(size):16 MB (16974309 bytes)
格式(extension):pdf
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Linear algebra is a fundamental tool in many fields, including mathematics and statistics, computer science, economics, and the physical and biological sciences. This undergraduate textbook offers a complete second course in linear algebra, tailored to help students transition from basic theory to advanced topics and applications. Concise chapters promote a focused progression through essential ideas, and contain many examples and illustrative graphics. In addition, each chapter contains a bullet list summarising important concepts, and the book includes over 600 exercises to aid the reader's understanding. Topics are derived and discussed in detail, including the singular value decomposition, the Jordan canonical form, the spectral theorem, the QR factorization, normal matrices, Hermitian matrices (of interest to physics students), and positive definite matrices (of interest to statistics students).
Table of contents :
Coverpage......Page 2
Half Title Page......Page 3
Title Page......Page 4
Copyright Page......Page 5
Dedication......Page 6
Contents......Page 7
Preface......Page 13
Notation......Page 18
0.1 Functions and Sets......Page 22
0.3 Matrices......Page 23
0.4 Systems of Linear Equations......Page 28
0.5 Determinants......Page 31
0.6 Mathematical Induction......Page 33
0.7 Polynomials......Page 35
0.8 Polynomials and Matrices......Page 37
0.9 Problems......Page 39
0.10 Some Important Concepts......Page 41
1.1 What is a Vector Space?......Page 43
1.2 Examples of Vector Spaces......Page 45
1.3 Subspaces......Page 47
1.4 Linear Combinations and Span......Page 50
1.5 Intersections, Sums, and Direct Sums of Subspaces......Page 53
1.6 Linear Dependence and Linear Independence......Page 55
1.7 Problems......Page 59
1.9 Some Important Concepts......Page 61
2.1 What is a Basis?......Page 62
2.2 Dimension......Page 65
2.3 Basis Representations and Linear Transformations......Page 71
2.4 Change of Basis and Similarity......Page 77
2.5 The Dimension Theorem......Page 84
2.6 Problems......Page 86
2.7 Some Important Concepts......Page 89
3.1 Row and Column Partitions......Page 90
3.2 Rank......Page 96
3.3 Block Partitions and Direct Sums......Page 100
3.4 Determinants of Block Matrices......Page 106
3.5 Commutators and Shoda’s Theorem......Page 109
3.6 Kronecker Products......Page 112
3.7 Problems......Page 115
3.8 Notes......Page 121
3.9 Some Important Concepts......Page 122
4.1 The Pythagorean Theorem......Page 123
4.2 The Law of Cosines......Page 124
4.3 Angles and Lengths in the Plane......Page 125
4.4 Inner Products......Page 128
4.5 The Norm Derived from an Inner Product......Page 132
4.6 Normed Vector Spaces......Page 139
4.7 Problems......Page 141
4.9 Some Important Concepts......Page 145
5.1 Orthonormal Systems......Page 147
5.2 Orthonormal Bases......Page 149
5.3 The Gram–Schmidt Process......Page 151
5.4 The Riesz Representation Theorem......Page 155
5.5 Basis Representations......Page 156
5.6 Adjoints of Linear Transformations and Matrices......Page 157
5.7 Parseval’s Identity and Bessel’s Inequality......Page 162
5.8 Fourier Series......Page 164
5.9 Problems......Page 168
5.10 Notes......Page 174
5.11 Some Important Concepts......Page 175
6.1 Isometries on an Inner Product Space......Page 176
6.2 Unitary Matrices......Page 178
6.3 Permutation Matrices......Page 183
6.4 Householder Matrices and Rank-1 Projections......Page 185
6.5 The QR Factorization......Page 191
6.6 Upper Hessenberg Matrices......Page 195
6.7 Problems......Page 197
6.8 Notes......Page 202
6.9 Some Important Concepts......Page 203
7.1 Orthogonal Complements......Page 204
7.2 The Minimum Norm Solution of a Consistent Linear System......Page 207
7.3 Orthogonal Projections......Page 211
7.4 Best Approximation......Page 217
7.5 A Least Squares Solution of an Inconsistent Linear System......Page 223
7.6 Invariant Subspaces......Page 227
7.7 Problems......Page 231
7.9 Some Important Concepts......Page 237
8.1 Eigenvalue–Eigenvector Pairs......Page 238
8.2 Every Square Matrix Has an Eigenvalue......Page 244
8.3 How Many Eigenvalues are There?......Page 247
8.4 Where are the Eigenvalues?......Page 252
8.5 Eigenvectors and Commuting Matrices......Page 260
8.6 Real Similarity of Real Matrices......Page 263
8.7 Problems......Page 264
8.9 Some Important Concepts......Page 268
9.1 The Characteristic Polynomial......Page 270
9.2 Algebraic Multiplicity......Page 272
9.3 Similarity and Eigenvalue Multiplicities......Page 275
9.4 Diagonalization and Eigenvalue Multiplicities......Page 276
9.5 The Functional Calculus for Diagonalizable Matrices......Page 280
9.6 Commutants......Page 283
9.7 The Eigenvalues of AB and BA......Page 284
9.8 Problems......Page 287
9.10 Some Important Concepts......Page 294
10.1 Schur’s Triangularization Theorem......Page 295
10.2 The Cayley–Hamilton Theorem......Page 297
10.3 The Minimal Polynomial......Page 300
10.4 Linear Matrix Equations and Block Diagonalization......Page 304
10.5 Commuting Matrices and Triangularization......Page 309
10.6 Eigenvalue Adjustments and the Google Matrix......Page 311
10.7 Problems......Page 312
10.9 Some Important Concepts......Page 318
11.1 Jordan Blocks and Jordan Matrices......Page 319
11.2 Existence of a Jordan Form......Page 323
11.3 Uniqueness of a Jordan Form......Page 327
11.4 The Jordan Canonical Form......Page 332
11.5 Differential Equations and the Jordan Canonical Form......Page 333
11.6 Convergent Matrices......Page 337
11.7 Power Bounded and Markov Matrices......Page 339
11.8 Similarity of a Matrix and its Transpose......Page 345
11.9 The Invertible Jordan Blocks of AB and BA......Page 346
11.10 Similarity of a Matrix and its Complex Conjugate......Page 348
11.11 Problems......Page 350
11.12 Notes......Page 359
11.13 Some Important Concepts......Page 360
12.1 Normal Matrices......Page 362
12.2 The Spectral Theorem......Page 365
12.3 The Defect from Normality......Page 369
12.4 The Fuglede–Putnam Theorem......Page 370
12.5 Circulant Matrices......Page 371
12.6 Some Special Classes of Normal Matrices......Page 373
12.7 Similarity of Normal and Other Diagonalizable Matrices......Page 377
12.8 Some Characterizations of Normality......Page 378
12.9 Spectral Resolutions......Page 379
12.10 Problems......Page 384
12.12 Some Important Concepts......Page 390
13.1 Positive Semidefinite Matrices......Page 392
13.2 The Square Root of a Positive Semidefinite Matrix......Page 400
13.3 The Cholesky Factorization......Page 404
13.4 Simultaneous Diagonalization of Quadratic Forms......Page 406
13.5 The Schur Product Theorem......Page 408
13.6 Problems......Page 412
13.7 Notes......Page 418
13.8 Some Important Concepts......Page 419
14.1 The Singular Value Decomposition......Page 420
14.2 The Compact Singular Value Decomposition......Page 425
14.3 The Polar Decomposition......Page 429
14.4 Problems......Page 435
14.6 Some Important Concepts......Page 439
15.1 Singular Values and Approximations......Page 440
15.2 The Spectral Norm......Page 443
15.3 Singular Values and Eigenvalues......Page 446
15.4 An Upper Bound for the Spectral Norm......Page 451
15.5 The Pseudoinverse......Page 452
15.6 The Spectral Condition Number......Page 456
15.7 Complex Symmetric Matrices......Page 460
15.8 Idempotent Matrices......Page 462
15.9 Problems......Page 464
15.11 Some Important Concepts......Page 470
16.1 The Rayleigh Quotient......Page 472
16.2 Eigenvalue Interlacing for Sums of Hermitian Matrices......Page 474
16.3 Eigenvalue Interlacing for Bordered Hermitian Matrices......Page 477
16.4 Sylvester’s Criterion......Page 482
16.5 Diagonal Entries and Eigenvalues of Hermitian Matrices......Page 483
16.6 *Congruence and Inertia of Hermitian Matrices......Page 484
16.7 Weyl’s Inequalities......Page 488
16.8 *Congruence and Inertia of Normal Matrices......Page 489
16.9 Problems......Page 493
16.10 Notes......Page 499
16.11 Some Important Concepts......Page 500
A.1 The Complex Number System......Page 502
A.2 Modulus, Argument, and Conjugation......Page 505
A.3 Polar Form of a Complex Number......Page 510
A.4 Problems......Page 514
References......Page 517
Index......Page 518

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