On Euclidean distance matrices

Balaji, R. ; Bapat, R. B. (2007) On Euclidean distance matrices Linear Algebra and its Applications, 424 (1). pp. 108-117. ISSN 0024-3795

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Official URL: http://www.sciencedirect.com/science/article/pii/S...

Related URL: http://dx.doi.org/10.1016/j.laa.2006.05.013


If A is a real symmetric matrix and P is an orthogonal projection onto a hyperplane, then we derive a formula for the Moore-Penrose inverse of PAP. As an application, we obtain a formula for the Moore-Penrose inverse of an Euclidean distance matrix (EDM) which generalizes formulae for the inverse of a EDM in the literature. To an invertible spherical EDM, we associate a Laplacian matrix (which we define as a positive semidefinite n × n matrix of rank n - 1 and with zero row sums) and prove some properties. Known results for distance matrices of trees are derived as special cases. In particular, we obtain a formula due to Graham and Lovàsz for the inverse of the distance matrix of a tree. It is shown that if D is a nonsingular EDM and L is the associated Laplacian, then D-1 - L is nonsingular and has a nonnegative inverse. Finally, infinitely divisible matrices are constructed using EDMs.

Item Type:Article
Source:Copyright of this article belongs to Elsevier Science.
Keywords:Distance Matrices; Laplacian; Trees; Infinitely Divisible Matrices
ID Code:81579
Deposited On:07 Feb 2012 05:11
Last Modified:07 Feb 2012 05:11

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