Ela Block Distance Matrices

In this paper, block distance matrices are introduced. Suppose F is a square block matrix in which each block is a symmetric matrix of some given order. If F is positive semidefinite, the block distance matrix D is defined as a matrix whose (i, j)-block is given by D ij = F ii +F jj −2F ij. When each block in F is 1 × 1 (i.e., a real number), D is a usual Euclidean distance matrix. Many interesting properties of Euclidean distance matrices to block distance matrices are extended in this paper. Finally, distance matrices of trees with matrix weights are investigated. 1. Introduction. In this paper, we introduce and investigate the properties of block distance matrices. This study is motivated by Euclidean distance matrices (EDM) which are a special class of nonnegative matrices with interesting properties and applications in molecular conformation problems in chemistry [3], electrical network problems [8] and multidimensional scaling in statistics [4]. An n × n matrix D = [d ij ] is called a Euclidean distance matrix if there exists a set of n vectors, say {x 1 , · · · , x n }, in a finite dimensional inner product space such that d ij = x i − x j 2. A symmetric matrix A = [a ij ] is called a Gram matrix if