【摘 要】
：
A symmetric and nonnegative matrix with zero diagonal elements is called a predistance matrix. An n×n predistance matrix D= (dij) is said to be an n×n Euc
【机 构】
：
IndianStatisticalInstitute,India
【出 处】
：
The 24th International Workshop on Matrices and Statistics(第
论文部分内容阅读
A symmetric and nonnegative matrix with zero diagonal elements is called a predistance matrix. An n×n predistance matrix D= (dij) is said to be an n×n Euclidean distance matrix (EDM), if there exist n points p1,… ,pn in a Euclideanspace Rr satisfying dij=||pi-pj||2(i,j=1,2,…,n).If the points p1,…,pn are on a hypersphere in Rr ,then D is said to be spherical. An EDM that is not spherical is called nonspherical. A necessary and sufficient condition for a predistance matrix D to be an EDM is that B =-1/2 PDP is positive semidefinite, where P=In-1/nJ, J being the matrix of all ones. We discuss various expressions for the inverse (when it exists) and the Moore-Penrose inverse of a Euclidean distance matrix (EDM) that are determined only by the positive semidefinite matrix B associated with the EDM. Both spherical and nonspherical EDMs are considered. A formula for the inverse of a principal submatrix of an EDM is also derived, whose expression uses the Schur complement of the Laplacian of the EDM. As an application, we obtain an expression for the terminal Wiener index of a tree. The talk is based on joint work with Balaji (Linear Algebra and Its Applications,2007) and Kurata (Linear Algebra and Its Applications,2014).
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