# Given a distance matrix is there an isometric embedding?

I have distance matrix $D$ that was calculated by some distance (non-Euclidean but satisfying distance requirements). Is there a set of points in some Euclidean space such that it generates matrix of Euclidean distances that is equal $D$?

I know that if $G=-HDH/2$ is p.d. where H is the centering matrix then such embedding exists. However, I don't have any information about $G$.

• Choose one of the points are the origin. Use the "polarization identity" to write find $(x,y)$ for any other points $x,y$ using the known distances $|x|^2$, $|y|^2$, $|x+y|^2$. The embedding problem can be solved iff the resulting Gram matrix is positive-definite. – Noam D. Elkies Sep 30 '16 at 4:24
• – Noam D. Elkies Sep 30 '16 at 4:25