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I am interested in diagonalizing a general $n \times n$ matrix with entries of the form \begin{equation} \frac{1}{|f_i-f_j|^p} \hspace{40px} 1 \le i,j \le n \end{equation} where $f_i,p \in \mathbb{R}$ for $1 \le i \le n$. Notably, the matrix is a symmetric Toeplitz matrix. Is it possible to find the exact eigenvalues and/or the eigenvectors of such a matrix? I would like to diagonalize the general form, but also solutions for special values of $\{f_i\}$ and/or $p$ would be interesting.

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