= Positive definiteness of the grounded nine-point Poisson stencil
{title2=$u^TAu=\sum w_{ij}(u_i-u_j)^2$}
On a rectangular grid with zero <Dirichlet boundary conditions>, the nine-point <Poisson equation> matrix has diagonal $10/3$, axial off-diagonal entries $-2/3$, and diagonal-neighbor entries $-1/6$. Its <quadratic form> is the sum of squared differences over internal links, weighted by $2/3$ or $1/6$, plus the weighted squares of values linked to the fixed zero boundary. The connected axial grid forces a zero <quadratic form> to have all entries zero. This proves <positive-definite matrix> in every ordering. The <Jacobi method> converges because its nonnegative iteration matrix has row sums at most one, with a strict deficit at boundary-adjacent vertices; a hypothetical unit-modulus eigenvector would propagate its maximum modulus to such a deficit row, which is impossible.
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