We say that a matrix $A\in M\_n(\mathbb{C})$ is a conditionaly negative definite matrix if it is hermitian and if for all complex numbers $c\_1,\ldots,c\_n$ such that $c\_1+\cdots +c\_n=0$ we have
$$
\sum\_{j,k=1}^{n}c\_j\overline{c\_k}a\_{jk}\leq 0.
$$


I'm interested by non-trivial families $(A\_n)\_{n\in \mathbb{N}}$ of concrete matrices where each $A\_n$ is a conditionaly negative definite matrix of $M\_n(\mathbb{C})$ such that all diagonal entries are null (important: the dimension of $A\_n$ grows with $n$). 


Has anyone seen this sort of thing in the literature?


I'm also interested by the related examples of non-trivial infinite conditionaly negative definite matrices (the definition of these is similar) such that all diagonal entries are null. 



references for families of conditionaly negative definite matrices 
Please have a look at: [my answer here](https://mathoverflow.net/questions/44777/how-to-characterize-real-square-matrices-a-such-that-vav-0-for-all-real-vec/44789#44789)---there you will find several references, from which you can gather a list of nontrivial cnd matrices (especially, the nontrivial ones that arise from cnd kernels, which are related to Hilbert space embeddable (isometrically) metrics).


