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[资源] Fundamentals of Matrix Analysis with Applications by E.B.Saff,A.D.Snider

Fundamentals of Matrix Analysis with Applications
by Edward Barry Saff,Arthur David Snider

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 Gauss-Jordan
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. Two-Point Boundary Value Problem, 118
C. Electrostatic Voltage, 119
D. Kirchhoff’s Laws, 120
E. Global Positioning Systems, 122
F. Fixed-Point 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 First-Order 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

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2016-04-29 16:48:06, 5.2 M