Matrix Methods

by Bronson; Costa

• ISBN13: 9780123744272
• ISBN10: 012374427X
• Edition: 3rd
• Format: Hardcover
• Copyright: 2008-09-04
• Publisher: Elsevier Science

Summary

Matrix Methods: Applied Linear Algebra, 3e, as a textbook, provides a unique and comprehensive balance between the theory and computation of matrices. The application of matrices is not just for mathematicians. The use by other disciplines has grown dramatically over the years in response to the rapid changes in technology. Matrix methods is the essence of linear algebra and is what is used to help physical scientists; chemists, physicists, engineers, statisticians, and economists solve real world problems. * Applications like Markov chains, graph theory and Leontief Models are placed in early chapters * Readability- The prerequisite for most of the material is a firm understanding of algebra * New chapters on Linear Programming and Markov Chains * Appendix referencing the use of technology, with special emphasis on computer algebra systems (CAS) MATLAB

Table of Contents

Preface	p. xi
About the Authors	p. xiii
Acknowledgments	p. xv
Matrices	p. 1
Basic Concepts	p. 1
Problems 1.1	p. 3
Operations	p. 6
Problems 1.2	p. 8
Matrix Multiplication	p. 9
Problems 1.3	p. 16
Special Matrices	p. 19
Problems 1.4	p. 23
Submatrices and Partitioning	p. 29
Problems 1.5	p. 32
Vectors	p. 33
Problems 1.6	p. 34
The Geometry of Vectors	p. 37
Problems 1.7	p. 41
Simultaneous Linear Equations	p. 43
Linear Systems	p. 43
Problems 2.1	p. 45
Solutions by Substitution	p. 50
Problems 2.2	p. 54
Gaussian Elimination	p. 54
Problems 2.3	p. 62
Pivoting Strategies	p. 65
Problems 2.4	p. 70
Linear Independence	p. 71
Problems 2.5	p. 76
Rank	p. 78
Problems 2.6	p. 83
Theory of Solutions	p. 84
Problems 2.7	p. 87
Final Comments on Chapter 2	p. 88
The Inverse	p. 93
Introduction	p. 93
Problems 3.1	p. 98
Calculating Inverses	p. 101
Problems 3.2	p. 106
Simultaneous Equations	p. 109
Problems 3.3	p. 111
Properties of the Inverse	p. 112
Problems 3.4	p. 114
LU Decomposition	p. 115
Problems 3.5	p. 121
Final Comments on Chapter 3	p. 124
An Introduction to Optimization	p. 127
Graphing Inequalities	p. 127
Problems 4.1	p. 130
Modeling with Inequalities	p. 131
Problems 4.2	p. 133
Solving Problems Using Linear Programming	p. 135
Problems 4.3	p. 140
An Introduction to The Simplex Method	p. 140
Problems 4.4	p. 147
Final Comments on Chapter 4	p. 147
Determinants	p. 149
Introduction	p. 149
Problems 5.1	p. 150
Expansion by Cofactors	p. 152
Problems 5.2	p. 155
Properties of Determinants	p. 157
Problems 5.3	p. 161
Pivotal Condensation	p. 163
Problems 5.4	p. 166
Inversion	p. 167
Problems 5.5	p. 169
Cramer's Rule	p. 170
Problems 5.6	p. 173
Final Comments on Chapter 5	p. 173
Eigenvalues and Eigenvectors	p. 177
Definitions	p. 177
Problems 6.1	p. 179
Eigenvalues	p. 180
Problems 6.2	p. 183
Eigenvectors	p. 184
Problems 6.3	p. 188
Properties of Eigenvalues and Eigenvectors	p. 190
Problems 6.4	p. 193
Linearly Independent Eigenvectors	p. 194
Problems 6.5	p. 200
Power Methods	p. 201
Problems 6.6	p. 211
Matrix Calculus	p. 213
Well-Defined Functions	p. 213
Prblems 7.1	p. 216
Cayley-Hamilton Theorem	p. 219
Problems 7.2	p. 221
Polynomials of Matrices-Distinct Eigenvalues	p. 222
Problems 7.3	p. 226
Polynomials of Matrices-General Case	p. 228
Problems 7.4	p. 232
Functions of a Matrix	p. 233
Problems 7.5	p. 236
The Function e[superscript At]	p. 238
Problems 7.6	p. 240
Complex Eigenvalues	p. 241
Problems 7.7	p. 244
Properties of e[superscript A]	p. 245
Problems 7.8	p. 247
Derivatives of a Matrix	p. 248
Problems 7.9	p. 253
Final Comments on Chapter 7	p. 254
Linear Differential Equations	p. 257
Fundamental Form	p. 257
Problems 8.1	p. 261
Reduction of an nth Order Equation	p. 263
Problems 8.2	p. 269
Reduction of a System	p. 269
Problems 8.3	p. 274
Solutions of Systems with Constant Coefficients	p. 275
Problems 8.4	p. 285
Solutions of Systems-General Case	p. 286
Problems 8.5	p. 294
Final Comments on Chapter 8	p. 295
Probability and Markov Chains	p. 297
Probability: An Informal Approach	p. 297
Problems 9.1	p. 300
Some Laws of Probability	p. 301
Problems 9.2	p. 304
Bernoulli Trials and Combinatorics	p. 305
Problems 9.3	p. 309
Modeling with Markov Chains: An Introduction	p. 310
Problems 9.4	p. 313
Final Comments on Chapter 9	p. 314
Real Inner Products and Least-Square	p. 315
Introduction	p. 315
Problems 10.1	p. 317
Orthonormal Vectors	p. 320
Problems 10.2	p. 325
Projections and QR-Decompositions	p. 327
Problems 10.3	p. 337
The QR-Algorithm	p. 339
Problems 10.4	p. 343
Least-Squares	p. 344
Problems 10.5	p. 352
A Word on Technology	p. 355
Answers and Hints to Selected Problems	p. 357
Index	p. 411
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