Al-Muraeb A. Linear Algebra for Localization. Algorithms, Use Cases,...2026

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Al-Muraeb A. Linear Algebra for Localization. Algorithms, Use Cases,...2026

Textbook in PDF format 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 Prerequisite Skills Needed to Use This Book: This book assumes a mid-level (whatever that means!) knowledge of and experience in: • Ubuntu OS • C++ language (including OOP) and IDE and a beginner-level (again, whatever that means!) for • linear algebra • localization • UML class diagrams Nevertheless, Appendix A lists suggested C++ learning resources, as well as resources of how to set the MS VS Code IDE to build, run, and debug the example code. As far as linear algebra concepts, suggested references are given in-place in this book, for further reading. For localization problem, a suggested reference for further reading is. Despite their abundance, suggested resources for UML diagrams are listed in Appendix A as well. Who Should Read This Book? • Graduate, as well as undergraduate, students of disciplines having the localization problem and applications under their umbrella (e.g., electrical and computer engineering, robotics engineering, and computer science). • x. In addition—providing good understanding and know-how through this book—algorithm and software engineers developing localization solutions—and any others made of linear algebraic models—in various industries, who may have not received focused courses of the subject in their academics or need a refresher with depth. For implementations in this book, the C++ language is selected for its dominance across the industrial board and its rich linear algebra libraries. In Chapter 1, Basic Matrix Operations, matrix orthogonality, matrix and vector norms, matrix invertibility, and orthogonal projections are presented. All of those topics are necessary for the following chapters. In Chapter 2, I present unitary and sparse matrices frequently used in localization problems. Using linear algebra representation will require simplifying (decomposing/factorizing) matrices using various algorithms. For that, orthogonal transformations (reflections and rotations) are utilized, and it is presented in Chapter 3. Chapter 4 details matrix factorization and the common factorization algorithms (except SVD, which deserves its own later chapter). Then, orthogonal projections and matrix psudoinverses are presented in Chapter 5. To measure the error in position estimation, covariance must be computed. Chapter 6 details the covariance topic. Building up to Chapter 7, where the especially important and dominant matrix factorization algorithm of SVD is detailed. Chapter 8 presents vitally important topics of Jacobian, Hessian, and Gradient, which are instrumental in solving localization non-linear least squares problems using iterative minimization algorithms. Now, how much information can one get from a set of observations (measurements) used in position estimation? The quantification of such information by FIM and the information assessment using CRLB are detailed in Chapter 9. Finally, Chapter 10 presents matrix block operations used in iterative algorithms solving localization least squares problem within time-step evolution utilizing Kalman filtering. Preface Introduction Basic Matrix Operations Special Matrices Orthogonal Transformations Matrix Factorization Orthogonal Projections and Psudoinverse Covariance Singular Value Decomposition Jacobian, Hessian, and Gradient Fisher Information Matrix and the Cramér–Rao Lower Bound Matrix Block Operations and Matrix Kernel Appendix A C++ Resources, Code Build, Code Run, and Code Debug Appendix B Case Study: Effect of Reference Points Locations on Cramér–Rao Lower Bound for Arbitrary Position Estimators