Consider this
For any ordering of the grid points, the matrix A of the system \begin{align*} u_{i-1, j}+u_{i+1, j}+u_{i, j-1}+u_{i, j+1}-4 u_{i, > j}=h^{2} f_{i, j}. \end{align*} is symmetric and negative definite.
The given proof is:
The above equation implies that if $a_{i j} \neq 0$ for $i \neq j$, then the $i$-th and $j$-th points of the grid $(p h, q h)$, are nearest neighbours. Hence $a_{i j} \neq 0$ implies $a_{i j}=a_{j > i}=1$, which proves the symmetry of $A$. Therefore $A$ has real eigenvalues and eigenvectors.
It remains to prove that all the eigenvalues are negative ...
I really do not understand this. I can picture the grid. However, surely I can label the nondiagonal entries with $-4$ and get some sort of contradiction? I think there is some structure to $A$ or something that I am missing. In regards to the argument, I have $0$ understanding what the author.
Could someone clarify this for me?