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.

  • $\begingroup$ You question introduces a family of matrices, but then doesn't ask anything about 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 16:48
  • $\begingroup$ I'm interested by complicated or interesting examples of conditionally negative definite matrices with zero diagonal entries. $\endgroup$ – BigBill Feb 3 '12 at 16:52
  • $\begingroup$ Also, looking in Bekka, de la Harpe and Valette's book on Kazhdan's property (T), appendix C, then "conditionally negative type" is those real matrices which satisfy your condition (for real numbers $(c_i)$) 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 use complex numbers? $\endgroup$ – Matthew Daws Feb 3 '12 at 16:56
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    $\begingroup$ You can read Berg "Harmonic analysis on Semigroups" for some information of the useful definition of my post (page 67, with complex numbers). $\endgroup$ – BigBill Feb 3 '12 at 17:07
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    $\begingroup$ Yep, that's a good reference! But doesn't Proposition 3.2 in that book basically answer your question-- it gives a Hilbert space representation (a bit more complicated than my comment before). That would surely give you lots of non-trivial examples...? $\endgroup$ – Matthew Daws Feb 3 '12 at 17:49

Please have a look at: my answer here---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).


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