Euclidean Distance Matrices and Their Applications in Rigidity Theory

1. Front Matter

   Pages i-xiv

   PDF

2. Mathematical Preliminaries

   • Abdo Y. Alfakih

   Pages 1-28

3. Positive Semidefinite Matrices

   • Abdo Y. Alfakih

   Pages 29-50

4. Euclidean Distance Matrices (EDMs)

   • Abdo Y. Alfakih

   Pages 51-87

5. Classes of EDMs

   • Abdo Y. Alfakih

   Pages 89-113

6. The Geometry of EDMs

   • Abdo Y. Alfakih

   Pages 115-120

7. The Eigenvalues of EDMs

   • Abdo Y. Alfakih

   Pages 121-143

8. The Entries of EDMs

   • Abdo Y. Alfakih

   Pages 145-161

9. EDM Completions and Bar Frameworks

   • Abdo Y. Alfakih

   Pages 163-184

10. Local and Infinitesimal Rigidities

    • Abdo Y. Alfakih

    Pages 185-210

11. Universal and Dimensional Rigidities

    • Abdo Y. Alfakih

    Pages 211-235

12. Back Matter

    Pages 237-251

    PDF

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 ourapproach. 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.

• 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

“This monograph is more than a standard text on matrices and rigidity theory. It is particularly important for providing the necessary information to mathematicians who are not experts in these areas. I really enjoyed the way how the topics are presented.” (Shing So, zbMATH 1422.15002, 2019)

• Department of Mathematics and Statistics, University of Windsor, Windsor, Canada

  Abdo Y. Alfakih

​Abdo Y. Alfakih is a Professor in the Department of Mathematics and Statistics at the University of Windsor. He received his PhD in Industrial and Operations Engineering at the University of Michigan. His research interests are in the areas of combinatorial optimization, semidefinite programming. His current work focuses on new approaches to the Graph Realization Problem and its relatives (bar and tensegrity framework rigidity, global rigidity, dimensional rigidity, universal rigidity etc) using Euclidean distance matrices, projected Gram matrices, Gale transform and semidefinite programming.

Policies and ethics