Question

What is the necessary and sufficient condition (if there is any) that $n$ orthonormal vectors $v_1,v_2,\cdots,v_n$ are eigenvectors of a Euclidean distance matrix. When $n=2$, the orthonormal vectors are easily charcterized, i.e., $(1/\sqrt{2}, 1/\sqrt{2})$ and $(1/\sqrt{2}, -1/\sqrt{2})$.

Answer 1

This isn't an answer, but it's too long for a comment. As you're maybe aware, a real $n \times n$ matrix $M$ is a Euclidean distance matrix if and only if the following conditions hold:

• $M_{ij} \geq 0$ for all $i, j$
• $M_{ii} = 0$ for all $i$
• $M$ is symmetric
• $M$ is conditionally negative definite, that is, $$ x^t M x \leq 0 $$ whenever $x \in \mathbb{R}^n$ with $\sum_i x_i = 0$.

This was shown in: I. J. Schoenberg, Metric spaces and positive definite functions, Transactions of the AMS 44 (1938), 522-536.