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.

the family. So I don't really understand if you are asking just about "conditionally negative definite matrices with zero diagonal entries", or something else...? $\endgroup$ – Matthew Daws Feb 3 '12 at 16:48realmatrices which satisfy your condition (forrealnumbers $(c_i)$) with already the condition that $a_{jj}=0$ for all $j$. Such things always arise as $a_{ij} = \|x_i-x_j\|^2$ where $(x_i)$ is a sequence in a real Hilbert space. So if you question really about the fact you wish to usecomplexnumbers? $\endgroup$ – Matthew Daws Feb 3 '12 at 16:56