Squared distance matrices of trees with matrix weights

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 $Δ$, is the $ns \times ns$ block matrix with $Δ_{ij}=d(i,j)^2$, 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 $Δ$ and find $Δ^{-1}$ under some conditions.

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