If I have a metric $d(\cdot,\cdot)$ on the set $\{1,\dots,n\}$, are there well-known necessary or sufficient conditions for the existence of a matrix norm $Q$ that induces that metric on the unit vectors $e\_1,\dots,e\_n$? That is, under what conditions can I find $Q\succeq0$ such that $$(e\_i-e\_j)^TQ(e\_i-e\_j) = d(i,j)^2$$ for all $i$ and $j$?



When does a finite metric induce a matrix norm?
Your quadratic form $Q$ is uniquely defined by $d$ on the hyperplane $H$ defined by $\sum x\_i=0$. Further, $Q|\_H\ge 0$ if and only if your metric space is isometric to a subset of a Eulcidean space.


