Alternating direction implicit iteration ($G=(\alpha I-Y{)}^{-1}(\alpha I+X)(\alpha I-X{)}^{-1}(\alpha I+Y)$)

For a rectangular Poisson equation discretization with commuting negative-definite directional matrices $X,Y$, two alternating implicit directional solves give the displayed iteration matrix. If their shared eigenvectors have eigenvalues $-{\xi }_p,-{\eta }_q$ with ${\xi }_p,{\eta }_q>0$, the iteration eigenvalues are $(\alpha -{\xi }_p)(\alpha -{\eta }_q)/[(\alpha +{\xi }_p)(\alpha +{\eta }_q)]$. For $\alpha >0$ their moduli are less than one, proving convergence. At $\alpha =0$ the composed iteration is the identity and the forcing terms cancel, so arbitrary initial errors persist. Strict positivity of the parameter is therefore essential for this splitting.