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Euclidean Distance Matrices and Their Applications in Rigidity Theory

  • Abdo Y.¬†Alfakih

Table of contents

  1. Front Matter
    Pages i-xiv
  2. Abdo Y. Alfakih
    Pages 1-28
  3. Abdo Y. Alfakih
    Pages 29-50
  4. Abdo Y. Alfakih
    Pages 51-87
  5. Abdo Y. Alfakih
    Pages 89-113
  6. Abdo Y. Alfakih
    Pages 115-120
  7. Abdo Y. Alfakih
    Pages 121-143
  8. Abdo Y. Alfakih
    Pages 145-161
  9. Abdo Y. Alfakih
    Pages 163-184
  10. Abdo Y. Alfakih
    Pages 185-210
  11. Abdo Y. Alfakih
    Pages 211-235
  12. Back Matter
    Pages 237-251

About this book

Introduction

This book offers a comprehensive and accessible exposition of Euclidean Distance Matrices (EDMs) and rigidity theory of bar-and-joint frameworks. It is based on the one-to-one correspondence between EDMs and projected Gram matrices. Accordingly the machinery of semidefinite programming is a common thread that runs throughout the book. As a result, two parallel approaches to rigidity theory are presented. The first is traditional and more intuitive approach that is based on a vector representation of point configuration. The second is based on a Gram matrix representation of point configuration. 

Euclidean Distance Matrices and Their Applications in Rigidity Theory begins by establishing the necessary background needed for the rest of the book. The focus of Chapter 1 is on pertinent results from matrix theory, graph theory and convexity theory, while Chapter 2 is devoted to positive semidefinite (PSD) matrices due to the key role these matrices play in our approach. Chapters 3 to 7 provide detailed studies of EDMs, and in particular their various characterizations, classes, eigenvalues and geometry. Chapter 8 serves as a transitional chapter between EDMs and rigidity theory. Chapters 9 and 10 cover local and universal rigidities of bar-and-joint frameworks. This book is self-contained and should be accessible to a wide audience including students and researchers in statistics, operations research, computational biochemistry, engineering, computer science and mathematics.

Keywords

Euclidean Distance Matrix Euclidean Distance Matrices EDMs Spherical EDMs Eigenvalues of EDMs Multidimensional Scaling Local Rigidity Distance Geometry Matrix Completion Problems Universal Rigidity Semidefinite Matrices

Authors and affiliations

  • Abdo Y.¬†Alfakih
    • 1
  1. 1.Department of Mathematics and StatisticsUniversity of WindsorWindsorCanada

Bibliographic information