Heavy-ball error propagation matrix
= Heavy-ball error propagation matrix
{title2=$M$}
For $e_k=x^*-x_k$,
$$
\binom{e_{k+1}}{e_k}
=
\begin{pmatrix}
(1+\beta)I-\alpha A&-\beta I\\
I&0
\end{pmatrix}
\binom{e_k}{e_{k-1}}.
$$
If $A$ is diagonal with entries $\lambda_i$, a coordinate permutation turns this matrix into blocks
$$
\begin{pmatrix}1+\beta-\alpha\lambda_i&-\beta\\1&0\end{pmatrix}.
$$