Block distance matrices

Abstract

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.