= Dissipative Cayley-transform contraction
{title2=$\|(I-hL/2)^{-1}(I+hL/2)\|_2\leq1$}
If $\operatorname{Re}\langle Lv,v\rangle\leq0$ in a finite-dimensional Hilbert space, $I-hL/2$ is invertible for every positive $h$. The relation $v_+-v_-=(h/2)L(v_++v_-)$ gives $\|v_+\|^2-\|v_-\|^2=(h/2)\operatorname{Re}\langle L(v_++v_-),v_++v_-\rangle\leq0$. Thus the <Crank--Nicolson method> contracts for such a linear system even when $L$ is a <non-normal matrix>. This is a direct energy result, not an unqualified inequality by a scalar rational function evaluated at a logarithmic norm.
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