On Cartesian product of Euclidean distance matrices

Authors

Ravindra B. Bapat, Indian Statistical Institute (Delhi Centre)
Hiroshi Kurata, The University of Tokyo

Article Type

Research Article

Publication Title

Linear Algebra and Its Applications

Abstract

If A∈Rm×m and B∈Rn×n, we define the product A⊘B as A⊘B=A⊗Jn+Jm⊗B, where ⊗ denotes the Kronecker product and Jn is the n×n matrix of all ones. We refer to this product as the Cartesian product of A and B since if D1 and D2 are the distance matrices of graphs G1 and G2 respectively, then D1⊘D2 is the distance matrix of the Cartesian product G1□G2. We study Cartesian products of Euclidean distance matrices (EDMs). We prove that if A and B are EDMs, then so is the product A⊘B. We show that if A is an EDM and U is symmetric, then A⊗U is an EDM if and only if U=cJn for some c. It is shown that for EDMs A and B, A⊘B is spherical if and only if both A and B are spherical. If A and B are EDMs, then we derive expressions for the rank and the Moore–Penrose inverse of A⊘B. In the final section we consider the product A⊘B for arbitrary matrices. For A∈Rm×m,B∈Rn×n, we show that all nonzero minors of A⊘B of order m+n−1 are equal. An explicit formula for a nonzero minor of order m+n−1 is proved. The result is shown to generalize the familiar fact that the determinant of the distance matrix of a tree on n vertices does not depend on the tree and is a function only of n.

First Page

135

Last Page

153

DOI

10.1016/j.laa.2018.10.008

Publication Date

2-1-2019

Recommended Citation

Bapat, Ravindra B. and Kurata, Hiroshi, "On Cartesian product of Euclidean distance matrices" (2019). Journal Articles. 971.
https://digitalcommons.isical.ac.in/journal-articles/971