A Second Course in Linear Algebra (Cambridge Mathematical Textbooks)

Cover......Page 1
Half-title......Page 3
Title page......Page 5
Copyright information......Page 6
Dedication......Page 7
Table of contents......Page 9
Preface......Page 15
Notation......Page 19
0.3 Matrices......Page 23
0.4 Systems of Linear Equations......Page 28
0.5 Determinants......Page 30
0.6 Mathematical Induction......Page 33
0.7 Polynomials......Page 34
0.8 Polynomials and Matrices......Page 36
0.9 Problems......Page 38
0.10 Some Important Concepts......Page 40
1.1 What is a Vector Space?......Page 41
1.2 Examples of Vector Spaces......Page 43
1.3 Subspaces......Page 44
1.4 Linear Combinations and Span......Page 46
1.5 Intersections, Sums, and Direct Sums of Subspaces......Page 48
1.6 Linear Dependence and Linear Independence......Page 50
1.7 Problems......Page 53
1.9 Some Important Concepts......Page 54
2.1 What is a Basis?......Page 55
2.2 Dimension......Page 57
2.3 Basis Representations and Linear Transformations......Page 62
2.4 Change of Basis and Similarity......Page 66
2.5 The Dimension Theorem......Page 72
2.6 Problems......Page 74
2.7 Some Important Concepts......Page 75
3.1 Row and Column Partitions......Page 76
3.2 Rank......Page 81
3.3 Block Partitions and Direct Sums......Page 85
3.4 Determinants of Block Matrices......Page 90
3.5 Commutators and Shoda’s Theorem......Page 93
3.6 Kronecker Products......Page 96
3.7 Problems......Page 98
3.9 Some Important Concepts......Page 103
4.2 The Law of Cosines......Page 104
4.3 Angles and Lengths in the Plane......Page 106
4.4 Inner Products......Page 109
4.5 The Norm Derived from an Inner Product......Page 112
4.6 Normed Vector Spaces......Page 118
4.7 Problems......Page 120
4.9 Some Important Concepts......Page 123
5.1 Orthonormal Systems......Page 124
5.2 Orthonormal Bases......Page 126
5.3 The Gram–Schmidt Process......Page 127
5.4 The Riesz Representation Theorem......Page 130
5.5 Basis Representations......Page 131
5.6 Adjoints of Linear Transformations and Matrices......Page 133
5.7 Parseval’s Identity and Bessel’s Inequality......Page 136
5.8 Fourier Series......Page 138
5.9 Problems......Page 142
5.11 Some Important Concepts......Page 146
6.1 Isometries on an Inner Product Space......Page 147
6.2 Unitary Matrices......Page 149
6.3 Permutation Matrices......Page 153
6.4 Householder Matrices and Rank-1 Projections......Page 155
6.5 The QR Factorization......Page 160
6.6 Upper Hessenberg Matrices......Page 164
6.7 Problems......Page 165
6.8 Notes......Page 169
6.9 Some Important Concepts......Page 170
7.1 Orthogonal Complements......Page 171
7.2 The Minimum Norm Solution of a Consistent Linear System......Page 174
7.3 Orthogonal Projections......Page 177
7.4 Best Approximation......Page 181
7.5 A Least Squares Solution of an Inconsistent Linear System......Page 186
7.6 Invariant Subspaces......Page 189
7.7 Problems......Page 192
7.9 Some Important Concepts......Page 196
8.1 Eigenvalue–Eigenvector Pairs......Page 197
8.2 Every Square Matrix Has an Eigenvalue......Page 202
8.3 How Many Eigenvalues are There?......Page 205
8.4 Where are the Eigenvalues?......Page 209
8.5 Eigenvectors and Commuting Matrices......Page 216
8.6 Real Similarity of Real Matrices......Page 218
8.7 Problems......Page 219
8.9 Some Important Concepts......Page 222
9.1 The Characteristic Polynomial......Page 223
9.2 Algebraic Multiplicity......Page 225
9.3 Similarity and Eigenvalue Multiplicities......Page 227
9.4 Diagonalization and Eigenvalue Multiplicities......Page 228
9.5 The Functional Calculus for Diagonalizable Matrices......Page 232
9.6 Commutants......Page 235
9.7 The Eigenvalues of AB and BA......Page 236
9.8 Problems......Page 238
9.10 Some Important Concepts......Page 243
10.1 Schur’s Triangularization Theorem......Page 245
10.2 The Cayley–Hamilton Theorem......Page 247
10.3 The Minimal Polynomial......Page 250
10.4 Linear Matrix Equations and Block Diagonalization......Page 253
10.5 Commuting Matrices and Triangularization......Page 257
10.6 Eigenvalue Adjustments and the Google Matrix......Page 258
10.7 Problems......Page 260
10.9 Some Important Concepts......Page 264
11.1 Jordan Blocks and Jordan Matrices......Page 265
11.2 Existence of a Jordan Form......Page 268
11.3 Uniqueness of a Jordan Form......Page 272
11.4 The Jordan Canonical Form......Page 277
11.5 Differential Equations and the Jordan Canonical Form......Page 278
11.6 Convergent Matrices......Page 281
11.7 Power Bounded and Markov Matrices......Page 283
11.8 Similarity of a Matrix and its Transpose......Page 287
11.9 The Invertible Jordan Blocks of AB and BA......Page 288
11.10 Similarity of a Matrix and its Complex Conjugate......Page 291
11.11 Problems......Page 292
11.12 Notes......Page 298
11.13 Some Important Concepts......Page 300
12.1 Normal Matrices......Page 301
12.2 The Spectral Theorem......Page 304
12.3 The Defect from Normality......Page 307
12.4 The Fuglede–Putnam Theorem......Page 308
12.5 Circulant Matrices......Page 309
12.6 Some Special Classes of Normal Matrices......Page 311
12.7 Similarity of Normal and Other Diagonalizable Matrices......Page 314
12.8 Some Characterizations of Normality......Page 315
12.9 Spectral Resolutions......Page 316
12.10 Problems......Page 320
12.12 Some Important Concepts......Page 324
13.1 Positive Semidefinite Matrices......Page 326
13.2 The Square Root of a Positive Semidefinite Matrix......Page 333
13.3 The Cholesky Factorization......Page 337
13.4 Simultaneous Diagonalization of Quadratic Forms......Page 339
13.5 The Schur Product Theorem......Page 341
13.6 Problems......Page 344
13.7 Notes......Page 348
13.8 Some Important Concepts......Page 349
14.1 The Singular Value Decomposition......Page 350
14.2 The Compact Singular Value Decomposition......Page 354
14.3 The Polar Decomposition......Page 358
14.4 Problems......Page 363
14.6 Some Important Concepts......Page 366
15.1 Singular Values and Approximations......Page 367
15.2 The Spectral Norm......Page 369
15.3 Singular Values and Eigenvalues......Page 372
15.4 An Upper Bound for the Spectral Norm......Page 376
15.5 The Pseudoinverse......Page 378
15.6 The Spectral Condition Number......Page 381
15.7 Complex Symmetric Matrices......Page 385
15.8 Idempotent Matrices......Page 387
15.9 Problems......Page 388
15.11 Some Important Concepts......Page 393
16.1 The Rayleigh Quotient......Page 394
16.2 Eigenvalue Interlacing for Sums of Hermitian Matrices......Page 396
16.3 Eigenvalue Interlacing for Bordered Hermitian Matrices......Page 399
16.4 Sylvester’s Criterion......Page 403
16.5 Diagonal Entries and Eigenvalues of Hermitian Matrices......Page 404
16.6 *Congruence and Inertia of Hermitian Matrices......Page 405
16.7 Weyl’s Inequalities......Page 408
16.8 *Congruence and Inertia of Normal Matrices......Page 410
16.9 Problems......Page 413
16.10 Notes......Page 418
16.11 Some Important Concepts......Page 419
A.1 The Complex Number System......Page 420
A.2 Modulus, Argument, and Conjugation......Page 423
A.3 Polar Form of a Complex Number......Page 427
A.4 Problems......Page 430
References......Page 432
Index......Page 433