Eigenvalue of the 'norm' matrix

Question

A week ago, I asked a question about the matrix composed with absolute difference of distinct number must have exactly only one positive eigenvalue. But I have no idea how to prove it with 'norm' matrix $A=(a_{ij})_{n\times n}$, where $$ a_{ij}=\|\xi_{i}-\xi_{j}\|_2. $$ Here $\xi_i\in\mathbb{R}^m$ are distinct vector and $\|\cdot\|_2$ is the Euclidean norm. Also I have verified it with MATLAB for $n=20$ and $m=5$, but the method used for $m=1$ failed. Any advice is welcome!

Answer 1

This matrix is known and is called a distance matrix see e.g. in Wikipedia section Distance matrix. The fact that such matrices have exactly one positive eigenvalue is discussed, for example, in the paper by E. Bogomolny et al. Distance matrices and isometric embeddings.

Distance matrices were also discussed in MSE, e.g., Distance matrix properties