= Energy-norm proof of convergence of SOR
{title2=$0<\omega<2\Longrightarrow\|C_\omega\|_A<1$}
Let $A=D+L+L^T$ be a real <symmetric positive-definite matrix>, with $D$ diagonal and $L$ strictly lower triangular. Put $B=D/\omega+L$ and $C_\omega=I-B^{-1}A$ for $\omega\ne0$. The <matrix norm induced by an energy norm> reduces the question to $T=A^{1/2}C_\omega A^{-1/2}$. Direct multiplication and $B+B^T-A=(2/\omega-1)D$ give
$$
T^TT=I-\left(\frac2\omega-1\right)A^{1/2}B^{-T}DB^{-1}A^{1/2}.
$$
The left side is <positive semidefinite>. The <matrix> following the scalar coefficient on the right is <positive-definite>, since $D$ has positive diagonal and $B^{-1}A^{1/2}$ is an <invertible matrix>. For $0<\omega<2$, every <eigenvalue> of $T^TT$ therefore lies in $[0,1)$. Hence its <spectral radius> is below one and $\|C_\omega\|_A=\|T\|_2<1$. Iteration errors consequently decay geometrically in the <energy norm>. The argument proves convergence from every initial <vector>, without requiring the original <iteration matrix> to be <symmetric>.
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