Euclidean Distance Matrices (EDMs)

  • Abdo Y. Alfakih


This chapter provides an introduction to Euclidean distance matrices (EDMs). Our primary focus is on various characterizations and basic properties of EDMs. The chapter also discusses methods to construct new EDMs from old ones, and presents some EDM necessary and sufficient inequalities. It also provides a discussion of the Cayley–Menger matrix and Schoenberg Transformations.


  1. 2.
    A.Y. Alfakih, On rigidity and realizability of weighted graphs. Linear Algebra Appl. 325, 57–70 (2001)MathSciNetCrossRefGoogle Scholar
  2. 6.
    A.Y. Alfakih, On dimensional rigidity of bar-and-joint frameworks. Discret. Appl. Math. 155, 1244–1253 (2007)MathSciNetCrossRefGoogle Scholar
  3. 12.
    A.Y. Alfakih, A remark on the Manhattan distance matrix of a rectangular grid, 2012. arXiv/1208.5150Google Scholar
  4. 19.
    A.Y. Alfakih, H. Wolkowicz, Some necessary and some sufficient trace inequalities for Euclidean distance matrices. Linear Multilinear Algebra 55, 499–506 (2007)MathSciNetCrossRefGoogle Scholar
  5. 21.
    A.Y. Alfakih, A. Khandani, H. Wolkowicz, Solving Euclidean distance matrix completion problems via semidefinite programming. Comput. Optim. Appl. 12, 13–30 (1999)MathSciNetCrossRefGoogle Scholar
  6. 24.
    S. AlHomidan, R. Fletcher, Hybrid methods for finding the nearest Euclidean distance matrix, in Recent Advances in Nonsmooth Optimization (World Scientific Publishing, River Edge, 1995), pp. 1–17CrossRefGoogle Scholar
  7. 25.
    S. AlHomidan, H. Wolkowicz, Approximate and exact completion problems for Euclidean distance matrices using semidefinite programming. Linear Algebra Appl. 406, 109–141 (2005)MathSciNetCrossRefGoogle Scholar
  8. 37.
    F.L. Bauer, C.T. Fike, Norms and exclusion theorems. Numer. Math. 2, 137–141 (1960)MathSciNetCrossRefGoogle Scholar
  9. 38.
    F. Bavaud, On the Schoenberg transformations in data analysis: theory and illustrations. J. Classif. 28, 297–314 (2011)MathSciNetCrossRefGoogle Scholar
  10. 40.
    J. Bénasséni, A variance inequality ensuring that a pre-distance matrix is Euclidean. Linear Algebra Appl. 416, 365–372 (2006)MathSciNetCrossRefGoogle Scholar
  11. 45.
    L.M. Blumenthal, Theory and Applications of Distance Geometry (Clarendon Press, Oxford, 1953)zbMATHGoogle Scholar
  12. 46.
    L.M. Blumenthal, B.E. Gillam, Distribution of points in n-space. Am. Math. Mon. 50, 181–185 (1943)MathSciNetCrossRefGoogle Scholar
  13. 52.
    A. Cayley, On a theorem in the geometry of position. Camb. Math. J. 2, 267–271 (1841)Google Scholar
  14. 53.
    Y. Chabrillac, J.-P. Crouzeix, Definiteness and semidefiniteness of quadratic forms revisited. Linear Algebra Appl. 63, 283–292 (1984)MathSciNetCrossRefGoogle Scholar
  15. 66.
    G.M. Crippen, T.F. Havel, Distance Geometry and Molecular Conformation (Wiley, New York, 1988)zbMATHGoogle Scholar
  16. 67.
    F. Critchley, On certain linear mappings between inner-product and squared distance matrices. Linear Algebra Appl. 105, 91–107 (1988)MathSciNetCrossRefGoogle Scholar
  17. 68.
    J.-P. Crouzeix, J.A. Ferland, Criteria for quasi-convexity and pseudo-convexity: relations and comparisons. Math. Program. 23, 193–205 (1982)CrossRefGoogle Scholar
  18. 70.
    M. Deza, M. Laurent, Geometry of Cuts and Metrics, Algorithms and Combinatorics, vol. 15 (Springer, Berlin, 1997)CrossRefGoogle Scholar
  19. 80.
    M. Fiedler, Elliptic matrices with zero diagonal. Linear Algebra Appl. 197/198, 337–347 (1994)MathSciNetCrossRefGoogle Scholar
  20. 84.
    D. Gale, Neighboring vertices on a convex polyhedron, in Linear Inequalities and Related System (Princeton University Press, Princeton, 1956), pp. 255–263zbMATHGoogle Scholar
  21. 92.
    J.C. Gower, Euclidean distance geometry. Math. Sci. 7, 1–14 (1982)MathSciNetzbMATHGoogle Scholar
  22. 93.
    J.C. Gower, Properties of Euclidean and non-Euclidean distance matrices. Linear Algebra Appl. 67, 81–97 (1985)MathSciNetCrossRefGoogle Scholar
  23. 99.
    B. Grünbaum, Convex Polytopes (Wiley, New York, 1967)zbMATHGoogle Scholar
  24. 102.
    T.L. Hayden, J. Wells, Approximation by matrices positive semidefinite on a subspace. Linear Algebra Appl. 109, 115–130 (1988)MathSciNetCrossRefGoogle Scholar
  25. 105.
    T.L. Hayden, R. Reams, J. Wells, Methods for constructing distance matrices and the inverse eigenvalue problem. Linear Algebra Appl. 295, 97–112 (1999)MathSciNetCrossRefGoogle Scholar
  26. 110.
    A.J. Hoffman, On the polynomial of a graph. Am. Math. Mon. 70, 30–36 (1963)CrossRefGoogle Scholar
  27. 117.
    G. Jaklič, J. Modic, A note on methods for constructing distance matrices and the inverse eigenvalue problem. Linear Algebra Appl. 437, 2781–2792 (2012)MathSciNetCrossRefGoogle Scholar
  28. 143.
    K. Menger, Untersuchungen uber allegemeine Metrik. Math. Ann. 100, 75–163 (1928)MathSciNetCrossRefGoogle Scholar
  29. 144.
    K. Menger, New foundation of Euclidean geometry. Am. J. Math. 53, 721–745 (1931)MathSciNetCrossRefGoogle Scholar
  30. 167.
    I.J. Schoenberg, Remarks to Maurice Fréchet’s article: Sur la définition axiomatique d’une classe d’espaces vectoriels distanciés applicables vectoriellement sur l’espace de Hilbert. Ann. Math. 36, 724–732 (1935)MathSciNetCrossRefGoogle Scholar
  31. 168.
    I.J. Schoenberg, On certain metric spaces arising from Euclidean spaces by a change of metric and their imbedding in Hilbert space. Ann. Math. 38, 787–793 (1937)MathSciNetCrossRefGoogle Scholar
  32. 169.
    I.J. Schoenberg, Metric spaces and completely monotone functions. Ann. Math. 39, 811–841 (1938)MathSciNetCrossRefGoogle Scholar
  33. 170.
    I.J. Schoenberg, Metric spaces and positive definite functions. Trans. Am. Math. Soc. 44, 522–536 (1938)MathSciNetCrossRefGoogle Scholar
  34. 196.
    H. Wolkowicz, G.P.H. Styan, Bounds for eigenvalues using traces. Linear Algebra Appl. 29, 471–506 (1980)MathSciNetCrossRefGoogle Scholar
  35. 197.
    H. Wolkowicz, G.P.H. Styan, More bounds for eigenvalues using traces. Linear Algebra Appl. 31, 1–17 (1980)MathSciNetCrossRefGoogle Scholar
  36. 200.
    G. Young, A.S. Householder, Discussion of a set of points in terms of their mutual distances. Psychometrika 3, 19–22 (1938)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

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

Personalised recommendations