Negative definiteness of a nine-point Dirichlet stencil ($u^{T}Au=-{\sum }_{\{v,w\}}{w}_{vw}(u_v-u_w{)}^2<0$)

Extend the interior grid vector by zero. With axial weights $2/(3h^2)$ and diagonal weights $1/(6h^2)$, the negative central coefficient equals the total neighbour weight. Expanding the edge-square sum gives the displayed quadratic form. Equality forces all axial differences to vanish, and connection to the zero exterior forces the vector to be zero. Simultaneous row and column permutations preserve symmetry and negative definiteness.