Matrix Methods (eBook)

Cover 1
Matrix Methods: Applied Linear Algebra 4
Copyright Page 5
TOC$Table of Contents 8
Preface 12
About the Authors 14
Acknowledgments 16
CH$Chapter 1. Matrices 18
1.1 Basic Concepts 18
Problems 1.1 20
1.2 Operations 23
Problems 1.2 25
1.3 Matrix Multiplication 26
Problems 1.3 33
1.4 Special Matrices 36
Problems 1.4 40
1.5 Submatrices and Partitioning 46
Problems 1.5 49
1.6 Vectors 50
Problems 1.6 51
1.7 The Geometry of Vectors 54
Problems 1.7 58
CH$Chapter 2. Simultaneous Linear Equations 60
2.1 Linear Systems 60
Problems 2.1 62
2.2 Solutions by Substitution 67
Problems 2.2 71
2.3 Gaussian Elimination 71
Problems 2.3 79
2.4 Pivoting Strategies 82
Problems 2.4 87
2.5 Linear Independence 88
Problems 2.5 93
2.6 Rank 95
Problems 2.6 100
2.7 Theory of Solutions 101
Problems 2.7 104
2.8 Final Comments on Chapter 2 105
CH$Chapter 3. The Inverse 110
3.1 Introduction 110
Problems 3.1 115
3.2 Calculating Inverses 118
Problems 3.2 123
3.3 Simultaneous Equations 126
Problems 3.3 128
3.4 Properties of the Inverse 129
Problems 3.4 131
3.5 LU Decomposition 132
Problems 3.5 138
3.6 Final Comments on Chapter 3 141
CH$Chapter 4. An Introduction to Optimization 144
4.1 Graphing Inequalities 144
Problems 4.1 147
4.2 Modeling with Inequalities 148
Problems 4.2 150
4.3 Solving Problems Using Linear Programming 152
Problems 4.3 157
4.4 An Introduction to the Simplex Method 157
Problems 4.4 164
4.5 Final Comments on Chapter 4 164
CH$Chapter 5. Determinants 166
5.1 Introduction 166
Problems 5.1 167
5.2 Expansion by Cofactors 169
Problems 5.2 172
5.3 Properties of Determinants 174
Problems 5.3 178
5.4 Pivotal Condensation 180
Problems 5.4 183
5.5 Inversion 184
Problems 5.5 186
5.6 Cramer’s Rule 187
Problems 5.6 190
5.7 Final Comments on Chapter 5 190
CH$Chapter 6. Eigenvalues and Eigenvectors 194
6.1 Definitions 194
Problems 6.1 196
6.2 Eigenvalues 197
Problems 6.2 200
6.3 Eigenvectors 201
Problems 6.3 205
6.4 Properties of Eigenvalues and Eigenvectors 207
Problems 6.4 210
6.5 Linearly Independent Eigenvectors 211
Problems 6.5 217
6.6 Power Methods 218
Problems 6.6 228
CH$Chapter 7. Matrix Calculus 230
7.1 Well-Defined Functions 230
Problems 7.1 233
7.2 Cayley–Hamilton Theorem 236
Problems 7.2 238
7.3 Polynomials of Matrices—Distinct Eigenvalues 239
Problems 7.3 243
7.4 Polynomials of Matrices—General Case 245
Problems 7.4 249
7.5 Functions of a Matrix 250
Problems 7.5 253
7.6 The Function eAt 255
Problems 7.6 257
7.7 Complex Eigenvalues 258
Problems 7.7 261
7.8 Properties of eA 262
Problems 7.8 264
7.9 Derivatives of a Matrix 265
Problems 7.9 270
7.10 Final Comments on Chapter 7 271
CH$Chapter 8. Linear Differential Equations 274
8.1 Fundamental Form 274
Problems 8.1 278
8.2 Reduction of an nth Order Equation 280
Problems 8.2 286
8.3 Reduction of a System 286
Problems 8.3 291
8.4 Solutions of Systems with Constant Coefficients 292
Problems 8.4 302
8.5 Solutions of Systems—General Case 303
Problem 8.5 311
8.6 Final Comments on Chapter 8 312
CH$Chapter 9. Probability and Markov Chains 314
9.1 Probability: An Informal Approach 314
Problems 9.1 317
9.2 Some Laws of Probability 318
Problems 9.2 321
9.3 Bernoulli Trials and Combinatorics 322
Problems 9.3 326
9.4 Modeling with Markov Chains: An Introduction 327
Problems 9.4 330
9.5 Final Comments on Chapter 9 331
CH$Chapter 10. Real Inner Products and Least-Square 332
10.1 Introduction 332
Problems 10.1 334
10.2 Orthonormal Vectors 337
Problems 10.2 342
10.3 Projections and QR-Decompositions 344
Problems 10.3 354
10.4 The QR-Algorithm 356
Problems 10.4 360
10.5 Least-Squares 361
Problems 10.5 369
Appendix: A Word on Technology 372
Answers and Hints to Selected Problems 374
IDX$Index 428