Squared distance matrices of trees with matrix weights

TL;DR

This paper derives formulas for the determinant and inverse of the squared distance matrix of a weighted tree with matrix weights, extending classical scalar results to matrix-weighted trees.

Contribution

It introduces new formulas for the determinant and inverse of squared distance matrices in trees with matrix weights, generalizing scalar cases.

Findings

Derived a formula for the determinant of the squared distance matrix.

Obtained an expression for the inverse of the squared distance matrix under certain conditions.

Extended classical scalar distance matrix results to matrix-weighted trees.

Abstract

Let $T$ be a tree on $n$ vertices whose edge weights are positive definite matrices of order $s$. The squared distance matrix of $T$, denoted by $\Delta$, is the $ns \times ns$ block matrix with $\Delta_{ij}=d(i,j)^2$, where $d(i,j)$ is the sum of the weights of the edges in the unique $(i,j)$-path. In this article, we obtain a formula for the determinant of $\Delta$ and find ${\Delta}^{-1}$ under some conditions.