metric property and psd matrix

Question

Let $X$ be a set, and $d : X × X \to R$ a metric on $X$ (which means that verifies : $d(x, y) \geq 0$ ,$d(x, y) = 0$ if and only if $x = y$ ,$d(x, y) = d(y, x)$ and $d(x, y) \leq d(x, z) + d(z, y)$),

My question is if the $n$x$n$ matrix defined by $$M_{ij}:=(1-d_{ij})$$with $d_{ij}=d(x_i,x_j)$ for $x_i$ and $x_j$ in $X$, is a positive semidefinite matrix?

Answer 1

No. Take $n=2, X=\mathbb R, d(x,y)=|x-y|, x_1=4$ and $x_2=1$. Then $M$ has the eigenvalue $-1$.Therefore $M$ is not positive semidefinite.