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 $(1/\sqrt{2}, 1/\sqrt{2})$.
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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:
This was shown in: I. J. Schoenberg, Metric spaces and positive definite functions, Transactions of the AMS 44 (1938), 522536. 

