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目录1 Vectors 1
1.1 Real vectors 4
1.2 Complex vectors 11
2 Matrices 15
2.1 Real matrices 19
2.2 Complex matrices 39
3 Vector spaces 43
3.1 Complex and real vector spaces 47
3.2 Inner-product space 61
3.3 Hilbert space 67
4 Rank, inverse, and determinant 73
4.1 Rank 75
4.2 Inverse 83
4.3 Determinant 87
5 Partitioned matrices 97
5.1 Basic results and multiplication relations 98
5.2 Inverses 103
5.3 Determinants 109
5.4 Rank (in)equalities 119
5.5 The sweep operator 126
6 Systems of equations 131
6.1 Elementary matrices 132
6.2 Echelon matrices 137
6.3 Gaussian elimination 143
6.4 Homogeneous equations 148
6.5 Nonhomogeneous equations 151
7 Eigenvalues, eigenvectors, and factorizations 155
7.1 Eigenvalues and eigenvectors 158
7.2 Symmetric matrices 175
7.3 Some results for triangular matrices 182
7.4 Schur’s decomposition theorem and its consequences 187
7.5 Jordan’s decomposition theorem 192
7.6 Jordan chains and generalized eigenvectors 201
8 Positive (semi)definite and idempotent matrices 209
8.1 Positive (semi)definite matrices 211
8.2 Partitioning and positive (semi)definite matrices 228
8.3 Idempotent matrices 231
9 Matrix functions 243
9.1 Simple functions 246
9.2 Jordan representation 255
9.3 Matrix-polynomial representation 265
10 Kronecker product, vec-operator, and Moore-Penrose inverse 273
10.1 The Kronecker product 274
10.2 The vec-operator 281
10.3 The Moore-Penrose inverse 284
10.4 Linear vector and matrix equations 292
10.5 The generalized inverse 295
11 Patterned matrices: commutation- and duplication matrix 299
11.1 The commutation matrix 300
11.2 The symmetrizer matrix 307
11.3 The vech-operator and the duplication matrix 311
11.4 Linear structures 318
12 Matrix inequalities 321
12.1 Cauchy-Schwarz type inequalities 322
12.2 Positive (semi)definite matrix inequalities 325
12.3 Inequalities derived from the Schur complement 341
12.4 Inequalities concerning eigenvalues 343
13 Matrix calculus 351
13.1 Basic properties of differentials 355
13.2 Scalar functions 356
13.3 Vector functions 360
13.4 Matrix functions 361
13.5 The inverse 364
13.6 Exponential and logarithm 368
13.7 The determinant 369
13.8 Jacobians 373
13.9 Sensitivity analysis in regression models 375
13.10 The Hessian matrix 378
13.11 Least squares and best linear unbiased estimation 382
13.12 Maximum likelihood estimation 387
13.13 Inequalities and equalities 391
Appendix A: Some mathematical tools 397
A.1 Some methods of indirect proof 397
A.2 Primer on complex numbers and polynomials 398
A.3 Series expansions 401
A.3.1 Sequences and limits 402
A.3.2 Convergence of series 403
A.3.3 Special series 404
A.3.4 Expansions of functions 407
A.3.5 Multiple series, products, and their relation 408
A.4 Further calculus 409
A.4.1 Linear difference equations 409
A.4.2 Convexity 410
A.4.3 Constrained optimization 410
Appendix B: Notation 415
B.1 Vectors and matrices 415
B.2 Mathematical symbols, functions, and operators 418
Bibliography 423
Index 426
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