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Given the $N \times N$ diagonal matrices $\mathbf{D}$ and $\mathbf{W}$ as defined below

$ \begin{split} \mathbf{DW} &= \left[ \begin{array}{cccc} \beta_{1} & 0 & \cdots & 0 \\ 0 & \beta_{2} & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \beta_{N} \\ \end{array} \right] % \left[ \begin{array}{cccc} M & \sum_{m=0}^{M-1}{\mbox{e}^{-i m \pi \theta_{1,2}}} & \cdots & \sum_{m=0}^{M-1}{\mbox{e}^{-i m \pi \theta_{1,N}}} \\ \sum_{m=0}^{M-1}{\mbox{e}^{-i m \pi \theta_{2,1}}} & M & \cdots & \sum_{m=0}^{M-1}{\mbox{e}^{-i m \pi \theta_{2,N}}}\\ \vdots & \vdots & \ddots & \vdots \\ \sum_{m=0}^{M-1}{\mbox{e}^{-i m \pi \theta_{N,1}}} & \sum_{m=0}^{M-1}{\mbox{e}^{-i m \pi \theta_{N,2}}} & \cdots & M \\ \end{array} \right] \end{split} $

where $\theta_{i,j} = \theta_{j,i}$ and $\theta_{i,j} = 0$ when $i = j$.

I would like to know if there is a closed form for \begin{equation} Tr\left[ (\mathbf{DW})^k \right], \end{equation} where $Tr[.]$ is the matrix trace operator and $k$ is an integer greater than or equal to 1.

So far I have been able to manually find up to $Tr\left[ (\mathbf{DW})^3 \right]$ but it gets too cumbersome for $k \geq 4$. But perhaps, hopefully, someone out there knows a closed form for this.

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