论文信息 - Distance matrices and isometric embeddings

Distance matrices and isometric embeddings

We review the relations between distance matrices and isometric embeddings and give simple proofs that distance matrices defined on euclidean and spherical spaces have all eigenvalues except one non-positive. Several generalizations are discussed.

[1] I. J. Schoenberg. On Certain Metric Spaces Arising From Euclidean Spaces by a Change of Metric and Their Imbedding in Hilbert Space , 1937 .

[2] A. Zee,et al. Spectra of euclidean random matrices , 1999 .

[3] Leonard M. Blumenthal,et al. Theory and applications of distance geometry , 1954 .

[4] F. Gantmakher,et al. Théorie des matrices , 1990 .

[5] H. Piaggio. An Introduction to the Geometry of N Dimensions , 1930, Nature.

[6] I. J. Schoenberg. Remarks to Maurice Frechet's Article ``Sur La Definition Axiomatique D'Une Classe D'Espace Distances Vectoriellement Applicable Sur L'Espace De Hilbert , 1935 .

[7] M. Stephanov,et al. Random Matrices , 2005, hep-ph/0509286.

[8] A. Ron,et al. Strictly positive definite functions on spheres in Euclidean spaces , 1994, Math. Comput..

[9] Michel Deza,et al. Geometry of cuts and metrics , 2009, Algorithms and combinatorics.

[10] O. Bohigas,et al. Spectral properties of distance matrices , 2003, nlin/0301044.

[11] J. Wells,et al. Embeddings and Extensions in Analysis , 1975 .

[12] T. MacRobert. Higher Transcendental Functions , 1955, Nature.

[13] A. Vershik. Distance matrices, random metrics and Urysohn space , 2002 .