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Although I am not strong in advanced group theory and coxeter groups, but some how I sense some similarity between the following matrix I obtained and the geometries related to coxeter groups ... I am asking either for a discussion of this matrix form; if there exists similar form to it in the classes of known matrices (other than the general hermitian operator categorization)...

or if there are keywords that can reduce my research time...

the matrix is

\begin{bmatrix} 1 & {\rm e}^{j (d_1 - d_2)} & {\rm e}^{j (d_1 - d_3)} & \ldots & {\rm e}^{j (d_1 - d_N)} \\ {\rm e}^{j (d_2 - d_1)} & 1 & {\rm e}^{j (d_2 - d_3)} & \ldots & {\rm e}^{j (d_2 - d_N)} \\ {\rm e}^{j (d_3 - d_1)} & {\rm e}^{j (d_3 - d_2)} & 1 & \ldots & {\rm e}^{j (d_3 - d_N)} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ {\rm e}^{j (d_N - d_1)} & {\rm e}^{j (d_N - d_2)} & {\rm e}^{j (d_N - d_3)} & \ldots & 1 \end{bmatrix}

best regards and thanks for help

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what I have come to conclude is that the elements of this matrix are the element of multiplication group E=(e exp (jtheta),*)

whose elements are exponentials of imaginary numbers. the identitity of the group is number 1 and in fact the arrangement of the matrix is exactly a Cayley table of the exponentials group E.

this implies that this matrix relates an N points symmetrical geometry structure (since the variables are all distance metrics)

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