Eigenspace of Euclidean distance matrix. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T03:05:47Z http://mathoverflow.net/feeds/question/72356 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/72356/eigenspace-of-euclidean-distance-matrix Eigenspace of Euclidean distance matrix. Sunni 2011-08-08T14:33:48Z 2011-08-08T14:55:13Z <p>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 <a href="http://en.wikipedia.org/wiki/Euclidean_distance_matrix" rel="nofollow">Euclidean distance matrix</a>. When $n=2$, the orthonormal vectors are easily charcterized, i.e., $(1/\sqrt{2}, 1/\sqrt{2})$ and $(1/\sqrt{2}, -1/\sqrt{2})$.</p> http://mathoverflow.net/questions/72356/eigenspace-of-euclidean-distance-matrix/72357#72357 Answer by Tom Leinster for Eigenspace of Euclidean distance matrix. Tom Leinster 2011-08-08T14:55:13Z 2011-08-08T14:55:13Z <p>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:</p> <ul> <li>$M_{ij} \geq 0$ for all $i, j$</li> <li>$M_{ii} = 0$ for all $i$</li> <li>$M$ is symmetric</li> <li>$M$ is <strong>conditionally negative definite</strong>, that is, $$x^t M x \leq 0$$ whenever $x \in \mathbb{R}^n$ with $\sum_i x_i = 0$. </li> </ul> <p>This was shown in: I. J. Schoenberg, Metric spaces and positive definite functions, Transactions of the AMS 44 (1938), 522-536.</p>