Linear Algebra for Localization: Algorithms, Use Cases, and C++ Implementations

Cover....1

Half Title....2

Title....4

Copyright....5

Contents....6

Preface....10

Acronyms and Abbreviations....14

Introduction....16

0.1 Localization and Its Applications Overview....16

0.2 C Implementations and Chapters Roadmap....17

1 Basic Matrix Operations....18

1.1 Really Basic Matrix Operations!....18

1.2 Orthogonality....21

1.3 Norm....22

1.3.1 Vector Norm....22

1.3.2 Matrix Norm....23

1.4 Invertibility....24

1.5 Orthogonal Projections....25

1.6 Use Cases in Localization....25

2 Special Matrices....28

2.1 Matrix Rank....28

2.1.1 Definition....28

2.1.2 Implementation....30

2.2 Unitary Matrix....31

2.2.1 Definition....31

2.2.2 Implementation....32

2.3 Sparse Matrix....35

2.3.1 Definition....35

2.3.2 Implementation....36

3 Orthogonal Transformations....39

3.1 Matrix Reflection and Rotation....39

3.1.1 Definition....39

3.1.2 Implementation....40

3.2 Householder Reflection....44

3.2.1 Definition....44

3.2.2 Implementation....46

3.3 Givens Rotations....49

3.3.1 Definition....49

3.3.2 Implementation....51

4 Matrix Factorization....55

4.1 QR Factorization....55

4.1.1 Definition....55

4.1.2 Implementation....61

4.2 LU Factorization....67

4.2.1 Definition....67

4.2.2 Implementation....70

4.3 Matrix Symmetry and Positive Definiteness....73

4.3.1 Definition....74

4.4 Cholesky Factorization....74

4.4.1 Definition....75

4.4.2 Implementation....77

5 Orthogonal Projections and Psudoinverse....83

5.1 Projections and Orthogonal Projections....83

5.1.1 Definition....83

5.1.2 Implementation....85

5.2 Matrix Pseudoinverse....86

5.2.1 Definition....86

5.2.2 Implementation....87

6 Covariance....89

6.1 Definition....89

6.2 Implementation....90

7 Singular Value Decomposition....95

7.1 Definition....95

7.1.1 Algorithms....97

7.2 Implementation....103

8 Jacobian, Hessian, and Gradient....110

8.1 Definition....110

8.1.1 Jacobian....110

8.1.2 Hessian....111

8.1.3 Gradient....112

8.2 Implementation....114

9 Fisher Information Matrix and the Cramr–Rao Lower Bound....117

9.1 Fisher Information Matrix....117

9.1.1 Definition....117

9.2 Cramr–Rao Lower Bound....121

9.3 Implementation....124

10 Matrix Block Operations and Matrix Kernel....128

10.1 Matrix Block Operations....128

10.1.1 Definition....128

10.1.2 Implementation....129

10.2 Matrix Kernel....133

10.2.1 Definition....133

10.2.2 Implementation....133

Appendix A C Resources, Code Build, Code Run, and Code Debug....136

Appendix B Case Study: Effect of Reference Points Locations on Cramr–Rao Lower Bound for Arbitrary Position Estimators....144

Index....160

This book emphasizes the vital role of linear algebraic models in solving localization problems, as well as many other problems in algorithms, data science, and artificial intelligence. Localization has multi-industrial applications, which this book attempts to address through linear algebraic approaches while using the dominant C++ programming language in those industries.

Features

• This book provides clear, illustrative descriptions of the main linear algebra topics and advanced algorithms in localization problems
• It provides C++ implementations available via the associated eResource repository, including detailed explanations, flowcharts, UML diagrams and text, and code run output
• It also provides case studies by the author for advanced topics in automotive applications