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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.

edited Feb 3 2012 at 21:39

asked Feb 3 2012 at 16:39

BigBill
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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...? – Matthew Daws Feb 3 2012 at 16:48

I'm interested by complicated or interesting examples of conditionally negative definite matrices with zero diagonal entries. – BigBill Feb 3 2012 at 16:52

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? – Matthew Daws Feb 3 2012 at 16:56

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You can read Berg "Harmonic analysis on Semigroups" for some information of the useful definition of my post (page 67, with complex numbers). – BigBill Feb 3 2012 at 17:07

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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...? – Matthew Daws Feb 3 2012 at 17:49

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S. Sra · Accepted Answer · 2012-02-03 17:03:56Z UTC

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).

references for families of conditionaly negative definite matrices

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