= Contractivity of split Crank-Nicolson diffusion
{title2=$\|C_A(h)C_B(h)\|_2\le1$}
For symmetric negative semidefinite matrices $A,B$, each factor $C_A(h)=(I-hA/2)^{-1}(I+hA/2)$ has norm at most one, since its eigenvalues are $(1+h\lambda/2)/(1-h\lambda/2)$ with $\lambda\le0$. The same holds for $B$, so their ordered product is a contraction for every $h\ge0$, whether or not $A,B$ commute. This proves unconditional discrete <L2 norm> stability of the split <Crank-Nicolson method> for directional diffusion. Its splitting error remains first order in general, because the product's quadratic defect is $h^2[A,B]/2$.
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