TITLE: Eigenspace of Euclidean distance matrix.
QUESTION [6 upvotes]: 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 $(1/\sqrt{2}, -1/\sqrt{2})$.

REPLY [6 votes]: 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.