= Gauss-Seidel energy decrease identity
{c}
{title2=$\|e_k\|_A^2-\|e_{k+1}\|_A^2=d_k^TDd_k$}
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 $Ae_k=(D+L)d_k$. Expanding the two squared <energy norms> yields $2d_k^TAe_k-d_k^TAd_k=d_k^TDd_k$. Every diagonal entry of $D$ is positive, so the error energy strictly decreases unless $d_k=0$. In that case $Ae_k=0$ and the exact solution has already been reached. This identity exhibits the method as a sequence of energy-reducing coordinate corrections.
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