I'm trying to understand thoroughly the proof of the Lovász's formula to calculate the hitting time $H(s, t)$ between two nodes $s$ and $t$ in a random walk on a graph. At page 13 it gets to the following equation:
(1) $(I - M)H = J - 2mD$
He argues that $(I - M)$ is singular. How did he conclude that?
After that, the author says that the following can be a solution for $H$ at (1):
(2) $X = (I - M + M^*)^{-1}(J - 2mD)$
Where does this matrix $X$ came from? I even tried substituting $H$ by $X$ at (1) and it doesn't make any sense.
Finally, he argues that by diagonalization $H$ is therefore equal to
(3) $H(s,t) = 2m \sum_{k=2}^{n} \frac{1}{1-\lambda_k} \left( \frac{v_{kt}^2}{d(t)} - \frac{v_{ks}v_{kt}}{\sqrt{d(s)d(t)}} \right)$
Which method of diagonalization is this?
Could anyone help me with this proof?
Lovász paper: https://cs.yale.edu/publications/techreports/tr1029.pdf