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Fundamentals of Matrix Analysis with Applications
Год издания: 2016
Автор: Edward Barry Saff, Arthur David Snider
Издательство: Wiley
ISBN: 9781118953655
Язык: Английский
Формат: PDF
Качество: Издательский макет или текст (eBook)
Интерактивное оглавление: Да
Количество страниц: 408
Описание: An accessible and clear introduction to linear algebra with a focus on matrices and engineering applications
Providing comprehensive coverage of matrix theory from a geometric and physical perspective, Fundamentals of Matrix Analysis with Applications describes the functionality of matrices and their ability to quantify and analyze many practical applications. Written by a highly qualified author team, the book presents tools for matrix analysis and is illustrated with extensive examples and software implementations.
Beginning with a detailed exposition and review of the Gauss elimination method, the authors maintain readers’ interest with refreshing discussions regarding the issues of operation counts, computer speed and precision, complex arithmetic formulations, parameterization of solutions, and the logical traps that dictate strict adherence to Gauss’s instructions. The book heralds matrix formulation both as notational shorthand and as a quantifier of physical operations such as rotations, projections, reflections, and the Gauss reductions. Inverses and eigenvectors are visualized first in an operator context before being addressed computationally. Least squares theory is expounded in all its manifestations including optimization, orthogonality, computational accuracy, and even function theory. Fundamentals of Matrix Analysis with Applications also features:
 Novel approaches employed to explicate the QR, singular value, Schur, and Jordan decompositions and their applications
 Coverage of the role of the matrix exponential in the solution of linear systems of differential equations with constant coefficients
 Chapterbychapter summaries, review problems, technical writing exercises, select solutions, and group projects to aid comprehension of the presented concepts
Fundamentals of Matrix Analysis with Applications is an excellent textbook for undergraduate courses in linear algebra and matrix theory for students majoring in mathematics, engineering, and science. The book is also an accessible goto reference for readers seeking clarification of the fine points of kinematics, circuit theory, control theory, computational statistics, and numerical algorithms.
Оглавление
CONTENTS
PREFACE ix
PART I INTRODUCTION: THREE EXAMPLES 1
1 Systems of Linear Algebraic Equations 5
1.1 Linear Algebraic Equations, 5
1.2 Matrix Representation of Linear Systems and the GaussJordan
Algorithm, 17
1.3 The Complete Gauss Elimination Algorithm, 27
1.4 Echelon Form and Rank, 38
1.5 Computational Considerations, 46
1.6 Summary, 55
2 Matrix Algebra 58
2.1 Matrix Multiplication, 58
2.2 Some Physical Applications of Matrix Operators, 69
2.3 The Inverse and the Transpose, 76
2.4 Determinants, 86
2.5 Three Important Determinant Rules, 100
2.6 Summary, 111
Group Projects for Part I
A. LU Factorization, 116
B. TwoPoint Boundary Value Problem, 118
C. Electrostatic Voltage, 119
D. Kirchhoff’s Laws, 120
E. Global Positioning Systems, 122
F. FixedPoint Methods, 123
PART II INTRODUCTION: THE STRUCTURE OF GENERAL
SOLUTIONS TO LINEAR ALGEBRAIC EQUATIONS 129
3 Vector Spaces 133
3.1 General Spaces, Subspaces, and Spans, 133
3.2 Linear Dependence, 142
3.3 Bases, Dimension, and Rank, 151
3.4 Summary, 164
4 Orthogonality 165
4.1 Orthogonal Vectors and the Gram–Schmidt Algorithm, 165
4.2 Orthogonal Matrices, 174
4.3 Least Squares, 180
4.4 Function Spaces, 190
4.5 Summary, 197
Group Projects for Part II
A. Rotations and Reflections, 201
B. Householder Reflectors, 201
C. Infinite Dimensional Matrices, 202
PART III INTRODUCTION: REFLECT ON THIS 205
5 Eigenvectors and Eigenvalues 209
5.1 Eigenvector Basics, 209
5.2 Calculating Eigenvalues and Eigenvectors, 217
5.3 Symmetric and Hermitian Matrices, 225
5.4 Summary, 232
6 Similarity 233
6.1 Similarity Transformations and Diagonalizability, 233
6.2 Principle Axes and Normal Modes, 244
6.3 Schur Decomposition and Its Implications, 257
6.4 The Singular Value Decomposition, 264
6.5 The Power Method and the QR Algorithm, 282
6.6 Summary, 290
7 Linear Systems of Differential Equations 293
7.1 FirstOrder Linear Systems, 293
7.2 The Matrix Exponential Function, 306
7.3 The Jordan Normal Form, 316
7.4 Matrix Exponentiation via Generalized Eigenvectors, 333
7.5 Summary, 339
Group Projects for Part III
A. Positive Definite Matrices, 342
B. Hessenberg Form, 343
C. Discrete Fourier Transform, 344
D. Construction of the SVD, 346
E. Total Least Squares, 348
F. Fibonacci Numbers, 350
ANSWERS TO ODD NUMBERED EXERCISES 351
INDEX 393
