Gauss-Seidel energy decrease identity ($\Vert e_k{\Vert }_A^2-\Vert e_{k+1}{\Vert }_A^2=d_k^{T}D{d}_k$)

For a real symmetric positive-definite matrix $A=D+L+L^T$, let $e_k$ be the Gauss-Seidel method error and $d_k=e_k-e_{k+1}$. Its recurrence gives $A{e}_k=(D+L)d_k$. Expanding the two squared energy norms yields $2d_k^{T}A{e}_k-d_k^{T}A{d}_k=d_k^{T}D{d}_k$. Every diagonal entry of $D$ is positive, so the error energy strictly decreases unless $d_k=0$. In that case $A{e}_k=0$ and the exact solution has already been reached. This identity exhibits the method as a sequence of energy-reducing coordinate corrections.