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This question is a little specific. I have read about graph theory and saw in one place the following transformation of an Laplacian matrix $L$:

\begin{eqnarray} ULW=\begin{bmatrix} -(l_{11}-l_{22})&-(l_{13}-l_{23})&\cdots&-(l_{1m}-l_{2m})\\ -(l_{11}-l_{32})&-(l_{13}-l_{33})&\cdots&-(l_{1m}-l_{3m})\\ \vdots&\vdots&\ddots&\vdots\\ -(l_{11}-l_{m2})&-(l_{13}-l_{m3})&\cdots&-(l_{1m}-l_{mm})\\ \end{bmatrix} \end{eqnarray}

Where,

\begin{eqnarray} W=\begin{bmatrix} 0_{m-1}^T\\ -I_{m-1} \end{bmatrix}, L=\begin{bmatrix} l_{11}&\cdots&l_{1m}\\ \vdots&\ddots&\vdots\\ l_{m1}&\cdots&l_{mm}\\ \end{bmatrix}, U=\begin{bmatrix} 1_{m-1}& -I_{m-1} \end{bmatrix}. \end{eqnarray}

What I would like to know is if the ULW matrix has restrictions to be diagonalized, and what would they be?

Thanks!

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