Euclidean and circum-Euclidean distance matrices: characterizations and interlacing property

Abstract

Motivated by the inverse formula of the distance matrix of a tree and the Moore–Penrose inverse of a circum-Euclidean distance matrix (CEDM), in this paper, we study a general real square matrix M whose Moore–Penrose inverse can be expressed as the sum of a Laplacian-like matrix L and a rank one matrix. In particular, for a symmetric hollow matrix M, under an assumption, we show that M is a Euclidean distance matrix if and only if L is positive semidefinite. Based on this, we obtain a new characterization for CEDMs involving their Moore–Penrose inverses. As an application, we show that the distance matrices of block graphs and odd-cycle-clique graphs are CEDMs. Finally, we establish an interlacing property between the eigenvalues of a Euclidean distance matrix M (including the singular case) and its associated Laplacian-like matrix L, which generalizes the interlacing property proved for the distance matrices of trees.