Heavy-ball error propagation matrix ($M$)

For $e_k=x^{\ast }-x_k$,If $A$ is diagonal with entries ${\lambda }_i$, a coordinate permutation turns this matrix into blocks

For $A=\mathrm{diag}(1,\gamma )$, $\alpha =1/\gamma$, and $\beta =(1-{\gamma }^{-1/2}{)}^2$, every error-propagation block has spectral radius at most $1-{\gamma }^{-1/2}$, and equality occurs.