Given a matrix G as a N*N symmetric matrix, in which
$$G_{ij} = \begin{cases} \sum_{j=1}^{N} \frac{1}{R_{ij}}, &\quad \mbox{if j = i} \\ -\frac{1}{R_{ij}}, &\quad \mbox{otherwise}\end{cases}$$
$R_{ii}$ and $R_{ij}$ are greater than $0$ and $R_{ii}>R_{ij}$. Matrix $G$ is a symmetric diagonal dominant matrix.
how to do the eigen-decompostion? to transfer this matrix to $G = W D W^{-1} $format, where D is a diagonal N*N matrix.
Since matrix G is a symmetric matrix, thus $W^{-1} = W^{T}$, where$ W^{T} $ means the transpose of $W$.
I tried that let $W = 1_{N \times N}$, $W$ is $N$*$N$ matrix with all the elements being $1$, but it seems not correct.
So, how to write a general term of $W$ toward this $N$*$N$ matrix? Thank you.