Description
Book SynopsisThis book provides comprehensive coverage of matrix theory from a geometric and physical perspective, and the authors address the functionality of matrices and their ability to illustrate and aid in many practical applications.
Trade Review"Providing comprehensive coverage of matrix theory from a geometric and physical perspective, the book 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." (Zentralblatt MATH 2016).
"This is a straightforward modern introduction to matrices..... a very well done text, probably most suitable for engineering students." (Mathematical Association of America 2016).
Table of ContentsPreface 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
