MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 $(1/\sqrt{2}, -1/\sqrt{2})$.

share|cite|improve this question

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.

share|cite|improve this answer
    
+1, Thanks, I know this fact. Do you think if it is possible to characterize the orthogonal matrix $P$ such that $P^TMP$ is diagonal? – Sunni Aug 8 '11 at 22:56
    
OK, sorry not to have added anything new. (Personally I find this fact of Schoenberg's rather surprising, in that the triangle inequality doesn't have to be mentioned.) I'm afraid I don't have any further thoughts. – Tom Leinster Aug 9 '11 at 0:22

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.